Tip 1
I find that I'm best online at games where I know the map well.
Committing the map layout to memory, knowing where the weapons are and the paths most people take and where the safe spots, open areas, and dead ends are key.
Avoid open areas and dead ends. Watch for your opponents to expose themselves when going for a weapon location, and know where you can retreat to if fire gets too heavy.
Tip 2
Different games call for different strategies, Halo 3 for instance rewards you for being aggressive and offensive, Rainbow Six on the other hand rewards you for playing it safe and being defensive.
Halo is mostly about combo kills, one weapon to drain their shield, another weapon to finish them off. For instance in close combat situations a shotgun followed by a melee is a deadly combo.
Rainbow Six is all about stealth and lining up a single perfect shot.
In general your strategy should change with the game, the level, your weapons, and the situation in front of you, in that order. Don't say "I'm going to try to grenade people this round" instead know the situation your in and use the strategy most befitting that situation.
Tip 3 know when to move and when to hide
it's harder to hit a moving target, it's even more difficult when the target keeps changing directions. if you're under fire, MOVE, and change directions randomly and take paths that your enemy wont expect. If you can jump do that too, 3-dimensions are more difficult to deal with than 2.
Similarly when attacking others, try to predict their movement and path, being able to launch a rocket or throw a grenade to where they're GOING to be is a sure fire kill.
Tip 4 go for easy prey, it might sound cheap but really it's about exploiting your opponent's weaknesses. Find players who follow a pattern, always go for the same weapon or always follow the same path and use that to your advantage. Habits die hard and if you know that ever time "halofan34" dies he runs for the plasma grenades be ready to open fire on the location of the plasma grenades.
In general, put yourself in your enemy's position, look at yourself like an enemy and try to find your own weaknesses and try to avoid doing those things. similarly try to predict what you might do if you were in your enemy's position.
controller skill will come with practice, but even someone with mediocre controller finesse can run with the pros if they have a good knowledge of the maps, the weapon capabilities, and strategy.