Learn the maps:
Know where the enemies are and where they are likely to pop out. Half of my kills in CoD4 are because Im already aiming my gun somewhere and an enemy just appears in my aiming reticule. Eachmap has certain hotspots, and bottlenecks where your more likely to encounter the badguys. If your game has a rader/map feature, learn how to use it, and USE it!
In CS, as soon as a round starts, you need to charge ahead and know exactly where the two teams will meet. If you run too far, you will get smoked. Run to just before that point and aim at the corner, or doorway, or wherever you know they are about to pop out of.
Follow your teamates instead of striking out on your own. Theres strength in numbers. Dont run directly behind them, but stay in the same area. You can get alot of kills by shooting people already in firefights with other team members. Or enemies already wounded from just finished firefights. Remember, there is no 'Kill Stealing' in team based games.
In Unreal Tournament, and Halo type games, its all about map control. That means getting to the good powerups first, and then being there when they respawn again. Figure out the time between respawns of good weapons and powerups and plan to be there each time they spawn. Many good Halo and UT players simply run the same route on a map over and over, getting all the good stuff. That way they are well armed whenever they encounter resistance. If you have a certain powerup it means your opponents dont.
Whenever possible, try to keep the high ground. Its much easier to kill people running around below you than it is for them to kill you.
If you get killed by someone in a certain area, approach that same area with caution next time. People are creatures of habit and will tend to stay in the same place, or move thru the same area, the same way, over and over. Take advantage of that. Flank the guy who killed you the next time around, or thow a grenade where he was, chances are he will be there again. Its okay to get killed, but dont get killed the same way over and over. If you get sniped, dont run thru that area again. Go another way and try to flank that sniper. I cant even count the number of times Ive run ahead at the start of a round of CS, or S&D in COD4, and killed some guy coming out of a doorway, and then proceeded to kill that same guy, in that same spot, every single round.
Learn your weapons:
If you see an enemy from across the map and all you have is a smg, dont shoot at him, all you will do is alert him to your presence and if he has a rifle you'll be dead.
If you run out of bullets and you know there are more enemies just around the corner, dont reload, switch to your pistol.
Learn where to throw grenades. Knowing the angles I can get grenade kills by throwing them where I know enemies are likely to be hiding out. I kill em, but never even see them.
In point blank encounters, raising up your weapons to look thru the sights will get you killed. Learn to shoot from the hip in CQC (close quarters combat). Melee attacks are also very effective in most games, often one hit kills.
If the enemy can see you, dont stand still while shooting at them. Your just making yourself a taget. The smg 'left-right' strafe dance is one of the most effective tactics in CoD4. Learn to strafe. And for UT and Halo, learn to circle strafe. Also mix it up a bit, dont be predictable and continue to strafe in the same direction, the enemy will lead you and you'll be dead.
Learn to love Flash and Stun type grenades. If you run into a room and get killed still holding onto these grenades, you doing it wrong. Unless your suddenly sniped, you should almost never die with grenades on you. Throw grenades into windows and doorways of buildings you are about to enter, or around corners, or onto higher ledges. If your game allows you to 'cook' your grenades, always do so. I dont even want my grenades to land and blow up, I like em to blow up in midair right over the target.
Know your controls:
Know how to control your game and what is possible in it. Learn how and where to run/sprint, and how to quickly jump over things and climb up onto/over stuff. Mess with your sensitivity until it feels right for you. If your crosshairs are flying all over the screen you may need to turn your sensitivity down.
UT has some of the best controls ever in a shooter imo. Theres a double jump, and a wall jump. By combining these two you can flip around the maps like a monkey ninja. Its much harder to shoot that then a guy simply running down a hallway. Learn to rocketjump and grenade jump as well.
Practice:
Even if you follow all the tips I just gave you, at the end of the day, you still need to place your crosshairs over an enemy and pull the trigger quicker than he does. This only comes with practice. Most online shooter players play ALOT, and whatever game your playing probably isnt the first shooter they've ever played. It may take hours, days, or even weeks of sucking before you start to get better. You have to keep at it. No one goes to the gym once and works out and expects to get muscles.
If you need to, load up a private game with just yourself and run around the maps figuring stuff out. I did this with CoD maps to figure out stuff like how to get onto seemingly inacessable rooftops, and other strange places. For UT, play by yourself and work out routes to get powerups. Figure out how to quickly jump to higher or lower elevation levels, this will help you lose pursuers when you play.
Getter better at FPS's does mean playing alot, and putting in the time, but you have to do it intelligently.