Maximizing Your Aim Assist Boost
On consoles, most FPS games (BF3 included) support or force "aim assisting" functions. In BF3, I believe this is an option you can toggle, and you might try turning it on if it's not already.
What this feature does is it helps you track targets easier. Your aim will pull in the direction of enemies, and as you sweep across enemies, your stick sensitivity will adjust automatically, making your crosshairs seem to "stick" to enemies.
Also, when aiming down the sights of a weapon, the aim assist will cause the camera to zoom in a manner that is beneficial to you - for example, if you've got your crosshairs near an enemy and you start to aim down sights, as the view zooms you'll be more on target than you were before. In many cases, you can use this to "snap aim" between several targets in a small zone by repeatedly going into and out of "aim down sights" mode.
Knowing these aspects of auto-aim systems, you can use them to your advantage. For instance, when I encounter someone in a console FPS, my first reflex is to start firing while I start holding the "aim down sights" button. My first few rounds might fire a bit wide, but as I start to aim down sights, my sights will track (due to the auto-aim) and my spread will decrease (due to ADS).
If you're entering an area and you think someone might be camping, you can often times sweep your crosshairs across an area, and they will slow down or stick to locations where enemies are, even if they're hidden by cover. Any time you're running by a bush where an enemy might be laying in wait, you can use this trick to flush them out.
In addition to maximizing your benefit from aim assist, you can also play with your sensitivity. Some people swear by very high or very low sensitivity, but changing your sensitivity is generally best done in small steps, as you'll need to get used to the difference.
At higher sensitivities, you'll be more likely to be able to react to threats from the side or rear, and faster at lining up your shots when changing direction. However, some people feel that it's not as easy to make small adjustments when you're just a bit off target, or while sniping. They also make controllers with adjustable resistance analog sticks, which I find useful. If you set your sensitivity high, and set the resistance high, then you can make small, precise movements by exerting a small force, or make large, reactionary movements by exerting a large force.
Another thing to consider is where your aim generally is when moving. I try to keep my crosshairs approximately at chest level for wherever I think an enemy is likely to appear. Weapons in these realistic shooters have a decent amount of kick per-shot, so you've got to allow for the fact that where your first round goes is not where your last round is likely to go. You can also fire in shorter bursts if you think you're likely to end up off-target.
Finally, if your reflexes aren't as solid as your enemy, try outsmarting instead of out-twitching them. You might have a 50/50 chance of getting a kill in a direct encounter, but your chances of success are much higher if you have the element of surprise. Explore the map, and find the "secret routes" and back doors. When people are fighting it out in the central area, flank around and surprise them!