There are 32 medals in this game, including multiple tiers for many of them (such as 3 kills in a row and 10 kills in a row give you different medals). To save space, I stacked the tiers together or merged a tier of different weapons. Your medals progress can be seen in the medals section of the main menus.
Double Kill - Killionaire : Get up to 10 kills in a row with less than 4 seconds between kills
Killing Spree - Untouchable : Get up to 20 kills in a row without getting hit
Rifle Spree/Pistol Spree/Grenade Spree : Get 3 kills in a row with the associated weapon without getting hit
Target Practice/Pistol Perfect/Cluster Bomb : Get 6 kills in a row with the associated weapon without getting hit
Contact/Gunslinger/Let it Rain : Get 10 kills in a row with the associated weapon without getting hit
*All weapons fall into either the Rifle or Pistol category, and only grenades count for grenade kills (Brute Shot is a rifle)
Splatter Spree-Sunday Driver : Get up to 10 kills in a row while in a vehicle without getting hit
Assassin : Melee an enemy in the back (backspace)
Sentry : Kill a vehicle with a turret
Anticipation : Kill an enemy with a charged shot (using a plasma pistol: hold your thumb down for a few shots, release, then hold again to charge)
Demolition : Destroy a structure with a grenade
Swiss Army : Kill 5 enemies with 5 different weapons
Wrecking Ball : Destroy an enemy vehicle with a vehicle of your own
Many of these can be farmed on the very first mission, if you're having trouble with them. Chances are you will get all but maybe one or two of them naturally in a playthrough. Also note that once you unlock the medal, you can back out of the mission and it will still count.