A lot of great tips have already been outlined above, but I will go over some main points that I utilized to make it through this game on Legendary, followed by some tips that are mission specific that I found to be effective.
*** Generic Tips ***
1. Long Range Weapons
Keep a long-range weapon in your inventory at ALL times. Whether it be a DMR, a Needle Rifle or a Magnum. This is not too hard, given the amount of ammo boxes and dead Marines for human weapons, and the Skirmishers/Elites that use the Needle Rifles.
--If you ask: "which is better?" I'd suggest the Needle rifle if at all possible. The main reason being is that it's 100% effective with headshots - 1 headshot to an unshielded enemy = kill. (With Brutes you need 1 or 2 extra bullets to remove the helmets on the infantry and Leaders, and the Heroes have energy shields as well.) The bonus feature with the Needle rifle that makes it somewhat superior to the DMR is that 3 needles into an unshielded enemy creates a supercombine explosion - which instantly kills the target regardless of rank, and gives very minor damage to anything that is standing within 2 feet of it. This is handy for when headshots aren't possible. Now, it is harder to find ammo for this gun, so if you want to rely on the supercombine be a bit more conservative when firing.
2. Secondary Weapons
A great secondary weapon to have is the Plasma Pistol. You might think "oh, that gun is super weak!" Well, it's not. Remember the Plasma Pistol from Halo CE? Well, it's like that. Really powerful. Not only can it disable shields and vehicles with an overcharge shot, it can kill a Grunt at any distance with 3 bolts, and is remarkably accurate. If you are patient enough, you can easily take out an Elite solely with the Plasma Pistol as long as you have good aim and a quick trigger finger.
However, when encountering large amounts of tough enemies, I try to get a Shotgun. Every shot fired is either a kill or it will stun an enemy long enough for you to get in a quick melee or another shot. (Or switch to your long-range weapon and go for a headshot).
If you can find a heavy weapon (such as rockets, fuel rod cannon, plasma cannon, laser, etc) - and you KNOW there is a use for it like vehicles to take out or LARGE clusters of enemies, do that but ditch it immediately once the ammo is depleted or it's usefulness has run out.
-- Another tip is if you traded a good weapon for a heavy weapon, and the HW is empty, memorize where you dropped your previous weapon. If you haven't progressed that far in the area, take the extra 30 seconds to run back and get it. This also works for ammo - you can still backtrack a bit to go to ammo crates and restock.
3. Take It Slow!
Legendary difficulty isn't about running into a group of enemies and shooting their faces off Rambo style. You will die in an instant. The best way to approach any group of enemies is to start with the smaller targets. The way I did it was to use a long range gun to take out Grunts first, followed by Jackals/Skirmishers, then the Elites or Brutes. The only exception to this rule was if a Skirmisher flanked me (which they will often do) and I will take it out first. If you do not fire upon the high rank targets, they usually will not advance on your position but rather fire at you from a distance. In actual fact, I found the Grunts and Jackals to be more menacing if they are left in a group rather than a single Elite or Brute.
Now, once you do that and all thats left is 1 or more Elites/Brutes, here is where patience is the key. Elites tend to flee when they get low on shields or health, so being in a position where you can see behind their cover is a plus. What I usually did was pick them off with the DMR or Needle rifle, firing slowly into their head when it was visible. Once an Elite's shield drops he gets stunned, so use that 1 second to fire one last headshot, and the enemy drops.
--Another tip (once the Grunts, etc have been cleared) is to take out the Elite/Brute that is holding the most lethal weapon. Examples: Fuel Rod Cannon, Plasma Launcher, Concussion Rifle. These weapons are HIGHLY effective against you, so the faster you clear them off the battlefield, the better.
[EDIT] an exception to the above is if you have a Sniper Rifle on you. Then, by all means target high rank enemies first.
Don't always rely on your allies. The Marines tend to die extremely fast, and even if you give them heavy weapons they still tend to do the same. Also they aren't all that bright - they will shoot you with rockets if you're in fist fight with an Elite, or snipe you/through you when too close to an enemy. Also, I've seen them shoot themselves with said heavy weapons. So my advice is: Don't bother. The best way to make use of the Marines (unfortunately) is to use them as bait/a distraction when facing multiple high rank targets or Hunters.
Occasionally you will get another Spartan(s) as an ally. They are slightly more reliable in the fact that they can't die. The trade-off is that you can't exchange guns with them. Now, they act the same as the Marines; meaning that if you are in close-combat with an enemy, they can and will shoot through you to kill them. Just be aware of this.
-- If you are facing a group of tough opponents, a good thing to try (when possible) is to move as close as you can to said enemies, and your Spartan ally with follow you, allowing the enemies to target him/her instead, and the Spartan will retaliate. They do quite a bit of damage against enemies, which will help you get the kill if they don't do it themselves.
4b. Allies & Vehicles
My number one rule is as follows: DO NOT LET THE FRIENDLY AI DRIVE THE VEHICLE THAT YOU ARE IN! This is insta-death. The friendly AI in this game (on Legendary in particular) have no concept of moderation or patience. If you hop on the turret of a Warthog and a Marine or Spartan hops behind the wheel, GET OUT as soon as possible. The allies will just drive you right into the centre of the biggest group of enemies on the field and you will get obliterated.
If you must, drive the vehicle yourself and have an ally use the gun. This also isn't always effective because the friendly AI tends to only fire when the enemy is very close to you, and then they usually only fire in short bursts.
This brings me to my next point:
Sometimes it's better to not use a vehicle, even if it's available. The reason being is that you can either find or are given sufficient power weapons to take them out manually, or you kind find great vantage points that allow you to take out the enemy vehicles at a distance.
If you feel a vehicle is needed, firstly follow my above rule (never let friendlies drive). Secondly, be patient. All enemies do much more damage to you and your vehicle so coming in close and fast to deal damage isn't usually effective.
Stay away from Ghosts if at all possible. Their guns are weak and hard to aim properly, especially when it's damaged (they fire in random directions). Gauss and Rocket Warthogs are good to have if you can get one. The Gauss is a 1 hit kill for most things, and a 2-3 hit kill for tanks.
--Tip: Let a Marine take the gun at first, and drive on the "fringes" of enemy clusters - at enough distance so enemy fire is limited, but close enough so the Marine can shoot. Chances are, the Marine will die. Take this opportunity to park the Warthog (or whatever you have) at a good vantage point and slowly snipe out the enemies with the gun at your leisure. If another ally happens to get behind the wheel, just hop out and kill him. If it's a Spartan you'll have to play a game of "Jump out, Jump In, shoot, Jump out, repeat"
6. Using Checkpoints Intelligently
What I've noticed about this game is once you clear out an area or are close to clearing it out, you can get checkpoints at regular intervals. This is extremely useful!
Example: You are fighting a large number of Brutes. You run out of ammo for your DMR but you remember that there was an ammo crate back at the entrance to the building. You go back, and restock. What you can do here is get to a good hiding spot, and wait. Usually a checkpoint will pop after a couple minutes - this way if you happen to die you don't have to re-kill all the enemies you took out, and have to backtrack again to get more ammo.
I've also noticed the further you backtrack, the more often a checkpoint will occur.
--Sometimes you get a checkpoint in a bad place. Such as you have ran out of ammo for your main weapon, and your backup will be of no use in the situation. You keep running out and dying over and over and you can't seem to figure out a way to get past this without restarting the whole mission and doing it again.
Don't fret! A simple way to fix this is to commit suicide repeatedly. The game's mechanics have a "failsafe" feature that will shunt you back to the previous checkpoint before the bad one. (like if you got a checkpoint when a Fuel Rod Cannon shot is inches from your face). After a certain amount of deaths in quick succession after the checkpoint loads (I think around 10) the failsafe kicks in and you will be pushed back to the previous checkpoint. This gives you a chance to learn from your mistake and get more ammo or position yourself better for next time.
7. Fights Aren't Always Necessary
There are a few rare instances where battles can be avoided by either taking a stealth approach, or sprinting past the group of enemies. I will outline ones that I've found below.
*** Mission Specific Tips ***
Here I will outline some tougher areas of some of the levels and explain tips I used to get past them.
I didn't have much trouble with this level, but one part near the end is a bit tough. When you get in the Falcon and fly to the relay outpost, there is a part there where you have to defend Kat while she wires a door to close. You don't actually have to fight here. Instead, clear a path to the door and get inside. All you have to do is sit in the farthest corner away from the door and let the other Spartans do the work. After a few minutes, the door will close. I never was hit by any grenades or Concussion shots that happened to make it through the door.
The main thing here is to find and use the Sniper whenever it's available. One of the civilians you have to save (but can't, because they die so fast) has one, so take that ASAP and pick off the Elites.
After you follow the riverbed to the spire thing, Hunters will appear. Be aware that it is possible for Jun to get knocked off the cliff. He does not spawn back up top. Restart the checkpoint immediately if you got one from BEFORE his cliff-dive. If you are unfortunate enough to have a checkpoint pop after he falls, just activate the failsafe. I've also been told that if you wait an obscenely long time (15 minutes) the door will open by itself without Jun's codes. But I have not proven that myself.
After you get through that gate, you will cross a bridge that's attached to a rock wall. There will be a Sniper Rifle in a hard case next to a rock on the bridge. TAKE THIS. I never noticed it until I backtracked for ammo, It will make it tremendously easy to take out the Spec Ops Elites with Focus Rifles.
Tip of the Spear
Keep the Warthog for as long as possible. Try to boot Kat out of it ASAP so she doesn't drive. Fire the turret at the AA gun where the gun connects to the base, and you can kill the AA gun without having to go inside. This takes a fair bit of shots though, so be aware. One thing I found was that I let Kat take the gun, and I drove slowly towards the Gun, and she took out most of the infantry that was guarding it. After that, I traded seats, and then hopped right out and took the drivers seat and sped off back to where we came from. She stayed up near the front firing at the Ghosts, and I was able to park the 'Hog farther away and take out the AA gun, the Sniper nest, and the Ghosts from across the entire area. After that I let her have the gun until we came across the Wraiths. I stayed back and picked off enemies, and she drove into the centre of the group and the Warthog exploded. She fought the Hunters and Wraiths by herself. I managed to kill an Elite with a Plasma Launcher ant took out both Wraiths with it. I used small arms to slowly take out the Hunters, and we got the Evac checkpoint before all the enemies were killed.
Long Night of Solace
All I really did here was I stayed sort of behind Anchor 9. Very few enemies came back there, and I let my allies fend for themselves. I'd then pop out occasionally and take out a couple Seraphs and Banshees, then retreat back behind the station. It worked well to keep me alive, since I've noticed it's actually hard to avoid enemy fire once they have a lock on you.
Once I was on the Corvette, I waited for the 2 Spec-Ops Elites to jump up into space, and took them out one by one. I had a Shotgun and a DMR at the time. Then I floated down onto the central support nearest the energy shield blocking the ship from space, and headshotted each other Elite one at a time.
I ran out of ammo for the DMR and was forced to use a Plasma Pistol until I picked up a Needle Rifle from a Spec Ops Elite. There is one room where there are Elites with Needle Rifles and Focus Rifles, and Grunts with Fuel Rod Cannons. Take out the Grunts first, and don't worry about the Engineers. The shields they provide are actually very minimal.
There will be a point where you have to defend an elevator. You can go outside and utilize the turrets that are set up, but I stayed inside. This keeps you safe from the Phantom's gun as well as the Plasma Turrets on the side. The Brute Heroes with the Gravity Hammers are your priority. A well placed grenade followed by a couple headshots does the job in no time. There is also an ammo crate down the central staircase. (IIRC... If not, its a dead Marine with either a DMR or Assault Rifle)
The hardest part of this level is towards the end where you have to activate anti-air defences. I still had the jetpack from earlier in the level, and I advise you to keep that. I had a DMR and an assortment of secondary weapons until I took a Fuel Rod Gun from a fallen Grunt. I used this and the Jetpack to take out both Wraiths. Getting into the central building to activate the guns was the hardest part of all. There are several Brute Heroes, and one has a Gravity hammer. There are also several Leaders and Infantry. It is nearly impossible to get up there. The Marines that re standing at the central staircase are useless. I don't know if it's a glitch in the game, but they don't shoot and enemies don't target them. Not only that, but Spirit dropships come by every so often and drop Ghosts behind you, so you can't even take out the Brutes slowly.
What I did was I ran to the right of the central stairs towards the high wall, and jetpacked straight up, and landed on the window sill where the button is. I pressed it and immediately ran off down the stairs. Mission complete.
That may take a few tries, as the enemy spawn locations are random, but there is ALWAYS a Brute Hero with a Hammer and one with a Fuel Rod Cannon, and 2 Leaders with Concussion rifles.
This mission isn't too terribly hard. After you clear the Hospital and have to go to the nightclub, there is an Easter Egg which can be unlocked that replaces the Hunters with dancing Grunts and Brutes. Someone above has already linked that video in their solution so I don't feel the need to relink it. However - 90% of those Grunts will have Fuel Rod Cannons, and there are around 20-25 Grunts in that room, and 5 Brute Infantry. You need to deactivate the jammer to complete the mission. When you do so, the music stops and everyone starts shooting at you. It is impossible to just sprint out of the room and survive. You WILL die.
You can choose to face 4 Hunters and a small handful of Grunts or do it using the Easter Egg. I used the Easter Egg method. What I did was I ran in an killed the Brute DJ (killing any Brutes or Grunts also stops the music) and ran out of the room. I slowly picked off the Grunts as they came up the stairs, and 'naded the ones that didn't. I took that opportunity to acquire a Fuel Rod Cannon with full ammo.
There will be a point where you need to kill Elites and Engineers on a roof. This is very difficult as there are groups of 6 Banshees attacking you at the same time. What I did was I flew to the top of Club Errera, and picked off the Engineers with the DMR. Then I flew down, took out the Banshees and quickly bombed the Elites before more Banshees spawned. I then landed at the club and pushed the Falcon off the edge and called for Evac, which gave me a brand new Falcon.
Once you go to kill the last Jammer, you can actually fly the Falcon into the lobby of the building through the back. I did this and eliminated all the enemies in the lobby very quickly. After the jammer was deactivated I ran AS FAST AS I COULD toward the elevators, killing maybe 2 Drones on the way.
After I got the Scorpion, I went very slowly. I took out targets as soon as they were visible, and focused on Banshees, Ghosts, and Turrets when I saw them. Again, you can take out the AA guns at a distance. I was able to keep the tank alive for the "Tank Beats Everything" achievement.
Once I got to the "Defend Halsey's Lab" part, I only activated the closest turret. I grabbed a Jet Pack (located on the far right hand side of the door, on the wall) and stole a Banshee. I focused on taking out the other Banshees first, followed by Wraiths. I then performed bombing runs on the enemies in front of the door when I could. I didn't die at all this way. There will be a point where a Phantom drops a pair of Ghosts. I landed on top of the door to the lab, dropped down and took the Laser that is in a hard case in front of the door. I used this to take out the Ghosts. The reason being the Ghosts are RIDICULOUSLY accurate in their fire and took me out 100% of the time even though I was flying high in the air - not to mention they are driven by Elites which requires more than 1 Banshee Bomb to kill them.
The Pillar of Autumn
Right as the mission started I managed to jump down and land on the Elite next to the overturned truck. That took out his shields while not damaging mine and I head-shotted him with the pistol for the kill. This is also where you can get the "If They Came To Hear Me Beg" achievement.
When you get to the Mongoose(s), there is a house with a rocket launcher and ammo inside. Take it, as it will be useful later. When driving the Mongoose, don't stop for anything. Once you get into the clearing with the Scarabs, it may seem safer by just driving straight through underneath the Scarab. It's not. You will likely fall off and get killed, also the Plasma Turrets on the Scarab can see you and do tremendous amounts of damage. Just follow the road and you will be fine. You will still take damage, but you shouldn't die as long as you don't slow down.
After I got across the bridge I used the Rocket Launcher on the Grunt methane thing which killed most of them instantly, and used the remaining rockets on the sniper tower and a group of Brutes. After all the infantry were dead, I took a plasma pistol and overcharged the Wraith. I head-shotted the gunner, boarded and melee'd the driver. This left the Wraith intact and driveable. I hopped in and used it to take out the Suicide Grunts and the skirmishers coming down the hill from the cave with ease.
Once inside the shipyards you are first confronted with many Elites with heavy weapons. I had a Needle Rifle and a shotgun, which I traded for a Laser that I found next to some dead Marines. I focused on the Elite General with the Fuel Rod using the Laser, followed by quick head shots. I did the same with all the Elites. Once you get to the centre wall, I went up the stairs and took another Laser located against one of the middle upstairs doorways leading out to the enemy (the "Boneyard") I used it to take out the Fuel Rod Turret and the Ghosts. I then used a Sniper Rife (against another doorway not too far away) to take out all of the Elites, and DMR'd the Grunts. (There is a DMR up there as well as on the ground against the huge centre wall at the only doorway through)
I went inside the building along the long catwalk and helped Emile kill the Majors that were inside. I then restocked on DMR ammo which is located in the little hallway leading into the room where the "core" is kept in Invasion. It's a weapon crate/locker.
Once I got to the Mac gun, I just crouched behind it on the staircase until Keys started yelling "SHOOT IT NOW!", and then I ran up, hopped in and fired. Mission complete.
*** If you have any other questions or concerns feel free to comment or PM me and I'll get back to you. ***
Good luck, and remember - be patient!