My skills are heavily based on halo 2 as I didn't really care for 3 and only tried 4 for a week. But I am generally good a FPS games so I think I can still help :P
Grenades are your best friend. Learn how to place them in tight spots, bounce them off walls to get around an angle etc.
Stick with team mates as much as possible. You will never get better if you are always alone and getting destroyed. Stick with a team mate and help support them.
Learning to strafe side to side while staying on target is a very valuable skill. If the game has split screen, feel free to spawn in a dummy player just to practice.
Learning the maps and where the power weapons locations are also very important. Even if you can not get to them, slowing your enemy down will help your team to secure them.
Learn the ins and outs of each class. Figure out for yourself what each pro and con for each one is. This goes for every weapon as well. The staples have always been BR and Magnum I believe, and I almost always see pro players equipped with these weapons (not counting power weapons)
Never be afraid to take cover if you are being shot at. If you are being shot at from behind, run with your head down to make it harder for them to get headshots on you (Only if being shot from the side to behind, never from the front.)
Shooter games like this are also a mind game of deceiving your enemy. Try to think of what moves they will make to get to you, and think of ways to counter it. This will come from practice a lot of the time.
I have an old halo 2 video lurking around I could find if you are interested. Hopefully some of this info will help!
