Instructions
Seek advice. There are plenty of seasoned players in the FPS community that are more than happy to share tips or even teach you in-game.
Watch other players. Play online and keep mental notes of how the better players behave. Specifically, see how they behave when they hit their targets. Do not worry if it costs you a few matches, because eventually you learn to emulate their strategies.
Use aiming mode. FPS games have interchangeable firing modes. Aiming mode changes your view so that you see directly through your weapon's targeting reticule. This dramatically increases your accuracy.
Choose the right weapon. Each weapon is designed differently. Some guns, for example, have scopes on them, while others do not. Weapons with scopes give a more accurate view of your enemy.
Find a safe position. Whether you are in single or multiplayer mode, find a safe place from which to shoot. This includes barricades, windows and walls. Not only does this protect you from enemy fire, but it reduces the stress of getting an accurate shot.
Shoot in front of the target. Unless your target is stationary, keeping a moving target in your reticule is extremely difficult and inefficient. Instead, aim slightly ahead of him in the direction of his movement and begin firing. Ultimately, he will run into your projectiles. This works best with rapid-firing weapons, like machine guns.
Practice often. Do not expect to master aiming strategies immediately. The only way to learn is through trial and error. Focus on learning, not winning.