When I play FPS games, I find that I can rarely kill people before they kill me; often a whole SMG clip doesn't take them down, while they hit me with a few pistol shots and I die.
I think this is just me being very poor at aiming. I go for the neck, so the recoil hits the head... How else can I kill people more often?
It depends on the game of course but here are some rules that you may find helpful:
Don't spray, make every bullet count.
For shotguns and sniper rifles it means that must be confident that you are going to hit your enemy when you shoot. At first it might become even harder to kill someone but it's a proper way to improve your aim.
For assault rifles it means that (again, depends on the game) you should shoot in short bursts making a tiny half-second breaks to decrease the spread.
Learn shooting techniques like drag-shot or twitch-shot
A good mouse and mouse sensitivity that suits you. I think majority (debatable I guess...) of professional gamers use rather low mouse sensitivity. They use the whole arm and the whole mouse carpet to turn 180 and use their wrist for precise movement. It is also important to have enough space to move your mouse. I feel extremely uncomfortable when there is something fragile on the same table where I sit...
Turn off mouse acceleration. Everywhere: in mouse drivers, in OS settings, in the game. Most people agree that it helps but there are a few guys who like to have it on.
Turn off vertical synchronization, tune your network settings. For source games cl_interp, cl_updaterate, cl_cmdrate etc can make a difference.
This is entirely dependent on the game, but learn whether your guns are hitscan or projectile.
Hitscan weapons have their bullets travel instantly; a hit or miss is calculated as soon as you pull the trigger. Don't rely on visual feedback alone to determine if a weapon is hitscan -- depending on how the bullets are animated, they may appear to be slower than they really are. The most important thing here is that you shouldn't lead with Hitscan weapons, as doing so will only make you miss.
Projectile weapons, on the other hand, have actual projectiles that need to hit the target to do damage, so you do want to lead your shots.
Take the time to practice all of these things by yourself, in a practice game or against bots
REMEMBER: where is the enemy? how many are there? do i have good cover? can they see me? will i get hit? where am i looking? 360 degrees? or tunnel vision? can i get a shot in? shoot to scare or kill?
position, safety, and awareness is infinitely more important than trying to "kill someone before they can kill you" - a common mistake to make especially if you are not thinking clearly
I made a video series with some serious tactical and training/practicing strategies to help you get better at shooting games. It's a 4-part training series about an hour in total.
This one is probably what you're most interested in...
Part 2: (practicing strategies)
If you like that video you should watch the rest because they will help you even more.
This answer is aimed at non-realistic FPSes. This info is designed to assist anyone playing games like anything from the Tribes franchise, Quake, Half-Life, Unreal Tournament, etc.
This one is in all caps, header font and fully-written because I can't emphasize it enough. In fact I'm frankly stunned it hasn't been mentioned already except in Chase's answer and he only listed it as number 3, because this one is universal. I've been playing shooters since Doom when I was five, and I can tell you that the one thing that is consistent across all shooters -- and this includes realistic/pseudo realistic non-arena shooters like Counter-Strike or Call of Duty -- is that map knowledge will allow you to dominate anyone that does not have map knowledge. Period, end of discussion.
The importance of knowing the map layout becomes even more prominent in the arena shooters I'm targeting with this answer, for a couple of very important reasons:
Knowing the map in shooters is the difference between coasting along the top of the map geometry at ~40 miles per hour grabbing every armor and weapon pickup along the way before everyone else can and walking around aimlessly looking for someone to shoot. The latter tends to end in loss.
2. Get good at spatial reasoning.
Arena shooters are also usually defined by their lack of hitscan (instant hit from the moment you press fire -- no projectile) weaponry. This adds another thing you have to get good at; leading your target. And your projectile's behavior is likely going to be very different depending on the weapon you use. This one is especially important in Tribes because of the combination of wide-open areas, breakneck speeds, and non-hitscan weapons -- new players sometimes have a hard time hitting anything at all in Tribes.
My advice to you is to pick one weapon and stick with it for a while. A common example is using the rocket launcher in Quake. The RL is a very versatile weapon because it has splash damage, which allows you to aim less precisely to score damage, and it can boost your mobility in a pinch -- the ubiquitous rocketjump. This makes it a weapon that is both good for newbies to pick up and highly rewarding to master. On the flipside, learning the railgun first might be a poor choice because it doesn't offer anything beyond being able to do good damage at long range, and it's also highly unforgiving when you miss (long refire time). Also, limiting yourself to one weapon whenever possible like this allows you to eschew having to learn everything at once, making the game more enjoyable while you learn it; as opposed to trying to learn every weapon at once and being terrible with all of them for a long time.
The other thing you should focus on learning when you're new to a shooter is the movement mechanics. Moving around affects your shot just as much as moving your crosshair, and it also doubles as your primary defensive measure. Fortunately, movement is usually a constant thing that doesn't change (notable exceptions include class-based team games like Team Fortress and Tribes -- in those cases I also highly suggest you stick with one class until you feel comfortable), so you only have to learn movement once to start applying your skill to each weapon.
3. Most importantly... try to have fun even when you're doing poorly. Or you'll never truly enjoy the game.
This sounds like an oxymoron if you were to apply it to most modern military shooters like Call of Duty, because when people get ahead in that sort of game, they really get ahead. Killstreak bonuses being used against you, spawning in a worse location, being spawn-camped by vehicles, even racking up money in Counter-Strike... these are all things that don't make a game fun for someone that's losing an FPS.
The easiest way to avoid this is to play team-based gametypes, with a focus on objectives instead of kills. Deathmatch being "the basic gametype" of most shooters is kind of a misnomer; I think that deathmatch is actually better described as the arena for experienced players who have achieved a high level of mastery with the fundamentals of the game like map knowledge, weapon usage, and movement mechanics. Games like CTF, attack-and-defend, king of the hill, etc. allow you to contribute to your team winning without being good at killing players. Additionally, they allow you to be helpful without knowing all of the map because objectives are usually clearly indicated on the HUD. Tribes is extremely strong in this regard, offering the entire home base defense/offense "side-objective" that anyone can pursue at any time to give their team an advantage.
Over time, you will become familiar with the fundamentals of the game, and as you enter intermediate player status you may crave "the kill," what many consider to be the purest form of competition between two players. You might find yourself stopping your pursuit of an objective just to have an intense 1v1 duel with another player. Then you can truly elevate your status to that of one who racks up ridiculous killstreaks and pulls off the occasional amazing trick-shot. Try to avoid having this attitude at the beginning of the game until you learn the basics.
Play a lot. Play often.
The only way to get better is to play. At first it may seem that you are not getting any progress. There isn't any tricks to getting better. You just need to play.
Deathmatch servers can be more efficient as you don't need to wait for the next round to train your aiming.
On the strategy side, you can watch some pro games. Watching good players playing can help you understand the high level strategy and overall idea of what is good to do.
It's also important that your sensitivity is perfect for you.
The absolute best advice I've gotten on aiming was aim with your fingers, not your hand. Don't move the mouse how you do when you're browsing the internet; Extending and curling your fingers to aim will allow you to make far more precise movements.
Fingers were made for dexterity after all.
Three tips that I have not seen explicitly mentioned yet so I thought I would pitch in:
That sums it up for what I have to offer outside of what has already been stated..I +1 the practice against bots as, at least in terms of CS:GO, I had never played an online match (this has changed) but never lost a 1v1 against my Gold Nova friend due to practice against the highest difficulty of bots.
Drink caffeinated drinks and rest beforehand if you're tired!
Use every advantage you can by learning the levels, predicting other players, laying traps or attacking when they're busy. Players that shoot first or more accurately are not always the ones that kill and shooting at the first enemy is not necessary the best plan, especially in games such as CS:S where players are so fragile.
If you're not good at killing, at least try to coordinate with your team mates. Good team work some times compensates bad aiming (or at least makes it less evident).
i made a video on just this hope it helps -
