You need to keep moving if you wanna frag someone; if you stand around, you're an easy target. NEVER STOP MOVING! Always run, don't walk, and never run in a straight line if you can help it - if you zigzag about the place, you'll be harder to hit.

When you get in a fight, move around a lot too. But more on fighting later. First we'll cover navigation.

While the first thing you should work on is improving your combat skills, getting around the level is important too. Why? Oh yeah, that's right - so you can get the big guns. You won't get many kills walking around in the same room (something close to camping in itself), so try and learn the level. Find out where you can escape to if you're losing a fight. Learn where you can and can not use certain weapons - for example, find a nice high spot from where you can chuck grenades, rather than in a hole in the ground. Find an open area for the rocket launcher.

When you're looking for a weapon, chances are some jerk will be trying to kill you. Run away. If you're armed with a pistol and he's got the Rocket launcher or nuke gun in unreal tourney, you're an idiot if you take him on. If he's managed to get and  keep that gun, he's good - most players try to take out the guy with the nuke gun rather than some punk with a pistol. When you run away, jump and strafe. He'll find it harder to hit you.

Strafing
Strafing means to move sideways. Set you keyboard up so that you can easily strafe at all times (especially when running). Strafing will make you 5 times harder to hit. So instead of just standing there and aiming at a target in front of you, learn to strafe around your target. keep them in your sights using the mouse to aim and fire. Mastering this is the ultimate in multiplayer game play, Its easy to do and it feels great.

Aim for the head
Aiming for the head makes no difference in Quake 1 however in Quake II, Quake 3, Half Life and Unreal Tournament it is a great strategy. It can cause between 2-3 times the damage. This means you can bring down the enemy quickly using less ammo and with less chance of  being hurt. One shot to the head should kill most players unless they are heavily armored.
Note: Ducking in my opinion isn't so wise. Your movement is hindered and those players who would normally aim for your torso are  now targeting your head.

Jumping
Jumping is useful, but not as useful as strafing. Try Jumping when someone shoots a rocket at you. Chances are they are aiming at your feet, if you jump you will be guaranteed to suffer less blast damage and can even jump over the rocket. Jumping can also confuse the opposition as most players try to predict what you are going to do. Jumping is just one more thing that they must now factor in
Note:  Those players who think they are really good may like to try strafing, jumping and aiming at the same time.

Predicting Players Movement
Start predicting the opposition movements. Predict where they are going to go next. If they are near health, ammo or weapons chances are they will try to strafe towards it. So hint 101 is to "AIM THERE". Always predict and your frag count will improve.  

Run
If you think you are going to be killed and you don't want to collect all your weapons again my advise is RUN!. Running backwards also helps as you can shoot at anyone who chases you. Remember to strafe while running, otherwise you might as well be standing still.
Note: Some games have a Teleporters. So If you think your not being targeted and have the time, try using your Teleporters to Teleport to somewhere that is both higher and safer.