There are 32 types of medals. You can see what ones you haven't earned from the Main Menu, on the bottom in the Records Section you should see a Medals option.
Double Kill, Triple Kill, Overkill, Killtacular, Killtrocity, Killamanjaro, Killtastrophe, Killpocalypse and Killionaire: Kill 2-10 Enemies within 4 seconds of each other. I found these fairly easy to get on mission A-5. You have to stay alive for about 2 minutes while waves of enemies attack, if you are lucky you can get to 10 kills on the turret but it all depends on how the enemies swarm you, it is also fairly easy to get if you go a bit suicidal with the Shotgun. You also have Incendiary grenades in this level, they are useful for setting up a zone instant death for a few seconds.
Killing Spree, Killing Frenzy, Running, Riot, Rampage, Untouchable: Kill 3, 6, 10, 15, 20 enemies without getting hit. Using experience to use the Sentry makes this slightly easier. Play carefully but stay on the offensive, use grenades to your advantage. I got an untouchable on mission A-5, but it's not as easy as a Killionaire.
Rifle Spree, Target Practice, Contact: Get 3, 6, 10 kills consecutively with a Rifle without getting hit. Mission A-1, once you play it a few times very easy place to get this.
Pistol Spree, Pistol Perfect, Gunslinger: Get 3, 6, 10 kills consecutively with a Pistol without getting hit. Once you've earned a Contact Medal redo A-1 with your pistol instead.
Grenade Spree, Cluster Bomb, Make it Rain: Get 3, 6, 10 kills consecutively with Grenades without getting hit. Mission A-1 again, You start with 3 grenades and there are 3 more near your spawn in a box to the right, giving you a total of 6. Try to hit at least 2 grunts with each and you will have this easily. Make sure to not kill with any other weapon while going for this, it must be 10 grenade kills in a row. This really frustrated me because I didn't realize and I kept stupidly killing a grunt here or there with my rifle.
Splatter Spree, Road Rage, Sunday Drive: Get 3, 6, 10 kills consecutively with a Vehicle without getting hit. Mission A-4 you start near a Wraith, use it through the level the main thing to worry about are other Wraiths just keep your distance and keep shooting. You do NOT need to run over enemies, any weapon of your vehicle counts.
Assassin: Kill an enemy with a melee attack from behind. Mission A-1 try to assassinate an enemy after they come out of a drop pod. I don't have much advice for this one I got it accidentally.
Sentry: Destroy an Enemy Vehicle with a Shade or Machine Gun Turret. At the end of Mission A-5, just before the timer runs out, Wraiths show up, use the Turrets to destroy one.
Anticipation: Kill an enemy with a charged shot Using a Plasma Pistol hold down Left and Right click for about 2-3 seconds and release both at the same time. If the shot hits a Grunt it should one shot him Sorry, I only know the Mouse and Keyboard controls, but if you do the Tutorial it will teach you how to do this.
Demolition: Destroy a Structure with a Grenade Mission A-2 when you get to the first objective throw a grenade at it.
Swiss Army: Kill 5 Enemies with 5 Different Weapons I also got this one by accident, just use a variety of weapons.
Wrecking Ball: Destroy a Vehicle with a Vehicle Mission A-4 will give you tons of chances to do this