These basic Halo tips are the foundation of a great player. If you can get these fundamentals down perfectly then you'll probably add a least 5 kills every game. I know these can get boring to read, but they are incredibly important. So get comfortable and even write some notes as you read.
Sensitivity-The most important aspect of the entire game. Get your sensitivity right, and you just added 3 kills to your score. A perfect sensitivity will allow to get head shots with ease, the key to winning battles. Your sensitivity should feel perfect to you. The targeting reticule should move where you wanted it to.
Here's another tip for you: Change your sensitivity settings mid-game if needed. Sensitivity is not a "one-size-fits-all" thing for all gametypes. This is the most important Halo 2 tip.
Weapon Selection- Try to have a long- to mid-range weapon and a short-range weapon. This is a flexible rule though. If your playing a big map, a sniper and a battle rifle would be a good choice. For small maps, double SMG and a shotgun is a pretty good combo, too.
Just try to rely on instinct to decide which weapons would be a good choice. Don't be static in choosing weapons. You don't always need a short- and long-range weapon.
Decide what you need at that moment.
Jumping- Jump whenever you can. Jumping during battles will help you avoid damage and get you a better shot at your enemy. Do NOT jump when you are getting sniped. Jumping slows you down, making you an easier target for the sniper.
Grenades- Throw grenades before you even fire a shot. Throwing a grenade in the middle of battle will waste time and you'll probably end up dead before you can release the grenade.
Remember, a well-placed grenade and a couple battle rifle shots is the easiest kill in the game.
Melee Attack- Bash whenever you can in close-range battles. Most of the time, the melee attack will finish off your enemy. Also, melee attacks do more damage in the air.
For more details and Halo 2 tips, go to Halo Insider.