Tip: Stick with your teammates
Just kidding...
Alright, first, try not to die many times because you give more chances to the enemies to win. Second, pay close attention to your environment, your radar and of course, your vision. Judge when to extend (exit your cover and attack an enemy when you have more chances of winning) or over-extend (exit your cover and attack while you have very low shields).
Also, don't be afraid to steal any kills; remember that it's free for all. If you're in an area where lots of enemies are gathered attacking each other, spam your nades or / and snipe them from a safe distance with a precicion weapon.
I hope these helped, it's been a while since I played Halo 3...
P.S. You can also search online for guides, some of them are really useful