Aiming for the head always helps if you've got a marksman weapon at long range. If you're in close, trying hitting them a few times with your gun and then go in for the kill with a melee. Also, cover always helps, no matter what range you're at from your enemy.
Elites: Use a plasma-based weapon to take out their shields then use a a bullet/spike/needle weapon to kill them.
Brutes: About 2 shots to the head with the DMR or pistol will knock of the helmet exposing the head for an easy kill. Needlers will kill a brute in seven shots and a Needle Rifle will kill a brute in three shots.
Brute Cheiftains: Use both methods above.