Getting used to using a mouse/keyboard was a huge one for me when I first moved into PC gaming. Back when I started, I was convinced that mice couldn't be useful, so I played through Half-Life using only a keyboard. When I started playing CS against actual people, I was worthless because I couldn't change the elevation of my aim. So I had to learn how to effectively aim with a mouse; huge improvement.
I'd suggest playing some single-player stuff just to build muscle memory. It doesn't matter what the game is, so long as you can have a similar mouse sensitivity. After a few minutes of shooting zombies in L4D2 single-player, I can jump into a CS server and get quicker flick headshots because my muscle memory is primed for quick, precise flicks of the mouse. It's a lot easier to build up in single-player, since you don't have to wait around to respawn after dying (which, let's face it, we all do, especially when we're just learning a game). Once you've got a few years of FPS gameplay under your belt, you'll realize that although the gameplay in various FPS games may be wildly different, the base mechanics are all virtually identical, so jumping into a new one is not terribly difficult.
And, as others mentioned, you're only as good as your knowledge of the map. Getting to know the maps is crucial. I don't think it's quite as important as developing fast aiming reflexes, personally, but it's a very close second. If you can find a place where you can catch your enemy unaware, you're going to have the advantage regardless of your reflexes.