The SINGLE MOST IMPORTANT thing to remember if you want to be the best at an FPS game, you MUST know the map inside and out. This takes time, but you absolutely need to know the best vantage points for your weapon of choice, which in turn will give you an idea of where the enemy will be also. If it's a game like CS:S where you play match after match, you need to know where your enemies will be at the beginning of the match, and where the surviving ones most likely will be going after the opening firefights. You need to know the quickest route to your objective also. You need to know the less traveled paths when you don't want to be caught. You need to learn enemy camping spots so you don't run by a dark corner without looking, or you don't run straight through that door with a sniper 80 feet down the hallway or across the empty lot.
It's all about knowing the map.