This is my first guide so I hope you can get some help out of my tips.
First off, in my opinion, I thought this was the hardest halo of any yet. Ammo is really scarce and it doesn't go far. Try to play thru the game to get a feel and layout of the levels. Legendary doens't change the levels at all, it just give you WAAAY more bad guys. For the first 6 levels I would play it on Normal to understand it, then play it immediatly after on Legendary so I knew what to expect.
1. Noob Combo - I know it seems like a no-brainer but really. Use it. DMR or Covenant Carbine paired with a plama pistol. Keep it and pair it anytime you can.
2. Patience....Patience. - I found myself playing this like multiplayer during the first few levels, running and gunning, and all that did was get me respawning. Slow down. Play from a distance. I found many times I would get a promethean knights shields down(you'll see a blue pop) only to have to retreat since I had no shield left. I know you'll have to start over, but it's better than respawing.
3. Headshots are a must - Grunts in one, Jackals in two(one on their weapon arm, then one in the head) and so on. Turn on Grunt Birthday party to help you tell if you're getting headshots. It really helped me. That's why you want to use tip #1. KEEP A RANGED WEAPON AT ALL TIMES!!
4. Stay back - Play from a distance and if possible let them come to you. Legendary difficulty has most of the AI searching you out, so you don't have to find them, they will find you. Peek over corners and around edges. DON'T CHARGE....they will kill you. While you're at a distance, take out the smallest first. Grunts, Jackal Snipers and any of the small Prometheans. In later levels take out the promethian watchers first. Nothing worse than having a knights shields gone only to have a watcher put up a shild for him.
5. Get comfortable with ditching weapons - Your noob combo won't last long, so find something else. Use whatever you can find. Often times I backtracked a couple of minutes to get the DMR ammo or needler i passed on the ground. Search and search and search. Ammo boxes are in the smallest and weirdest of spots so look around. The needler is REALLY powerful so take advantage of it. Use your grenades and armor ablilites as well, that's why you got them. I found myself most of the time using hologram and promethian shield. Make good use of it. The AI is overly aggressive so throw out or hologram and hit them in the back.
6. Run - Don't be afraid to run. If all else fails and you've tried different methods, then just run. Many times I would throw out the hologram in the opposite direction and run the other way. In the level midnight, I did it a couple of times to get in the portals. Really helped me get past some crappy checkpoints.
7. Edit - Some of the comments make great suggestions. Anytime you know you have a pair of hunters coming save a Fuel Rod/Rocket Launcher. Most of the time thru my Legendary playthrough an Elite would be right in front of a pair of Hunters with a Fuel Rod or Rocket Launcher. Get it and save it. Very Useful. Also make good use of turrets. Take one with you for the road ahead. Saves ammo and knocks out Elites fast!
Some general level tips(minor spoilers here):
Requiem - While searching the crashed ships, climb all the way to the top on the outside of the 1st UNSC ship. Up there is a battle rifle and Sniper rifle. You can make good use out of that.
Forerunner - Noob Combo and hologram. I found it really useful on the Knights. Later in the level you'll have to knock out 3 forerunner generators with some banshees fling over head. As soon as you exit a big door to the left you'll see an Elite running for a banshee, kill him but leave the banshee. They will kill you fast if you try it. Here is where hologram comes in handy. Work your way around the outside and be patient. This part took me a while with many deaths.
Infinity - One of the harderst levels in the game, I thought. Towards the middle of the level you'll come to 3 or 4 marines at the bottom of the hill and a few weapons next to them. Leave those weapons there. You're going to need the ammo for the Promethean assault. Go all the way over to the right side of the screen and you'll see a big tree you can get behind. Use it for cover. Now since you left the weapons from before, you have extra ammo right below you. Don't worry about them coming from behind you, none ever did for me. They all climbed the hill and came at me head on.
Edit: Reclaimer - During the various assaults on the mammoth, get the Gauss warthog. I think it will be driving around while you ride in the mammoth. If you use it at all, bring it back and park it in the bottom, you're going to need it later. An A.I. will automatically jump in and run the cannon for you, as long as you drive. If you can manage to keep him alive, center him and the hog in the doorway and he will do all the work on the wraiths and ghosts that will come your way. I seem to remember the mammoth parks several times and one time in particular, it parks sideways(for whatever reason). Put the warthog and marine right over the open doorway, and just sit back and allow the A.I. to actually help. If he dies, just restart the checkpoint, but it should get you thru the various assault parts of the level. I only had to restart twice on this level, with this method.
Shutdown - Noob combo again. I also played this level from a distance and went slow.
Midnight - Hologram was a must for me. Towards the middle of the level you'll get to a Promethean Armory where you can pick any weapons. I took a Scattershot and an Incineration Cannon. A scattershot will kill a Knight in 3 hits and the Cannon is a instant kill. Save the cannon for the last part of the level when you are around the Didact with the knights, this is very helpful. As for the knights throw the hologram out then run behind them.
Hopefully any of these tips will help you. Again this is my first guide, so if have any tips let me know in the comments.