I admit, I stopped being an uber FPSer back in the late 90's, but I ruled the day in the era of Quake II and Jedi Knight (on 28k dial-up no less). I don't imagine the rules have changed too terribly since then.
I would say the best advice, as trite as it may sound, is just to practice. Get a good feel for the engine, so it becomes an extension of your natural movement. You can learn how to move fluidly in-game as if you were moving around in it yourself. Learn the quirks so you can compensate and adjust. Certain weapons for certain situations, off-kelter hitboxes, recoil, range of movement when jumping, awareness of shadows or sounds, work out your hand-eye co-ordination. The better you immerse yourself and the more natural you feel with the game, the better you'll be.
Second, I would say, learn the maps. Obviously, if you know your terrain, you'll do better. Especially against someone who doesn't know it so well. There's obviously a lot more to it than simply knowing the layout. Know the best places to go under different circumstances. Though I believe that knowing a map is critical to success, I put it second because if both/most players are on unfamiliar territory, whoever simply plays best within the game will likely win.
Third, watch and learn. Be observant. Adapt techniques that you see and evolve your own style. Don't condemn yourself to trying one technique over and over if it's not going so well. As far as opposing players, a lot of newbies are stupid. They'll make the same runs over and over again, or try the same stupid stuff like going into a certain area, jumping and shooting without moving around, then trying to hide behind a box or something. The style is easily compensated for and he either adjusts and gets better, or you rack up easy kills. Usually, you just rack up easy kills. Even more experienced players will get into patterns. Some players get so predictable that you can start firing blidly at a corner and kill him right as he comes around it because you saw him grab a certain item or power up just a few seconds ago, fitting their pattern. But don't fall victim to falling into patterns yourself. Mix it up as you go so you aren't predictable.