One of the earliest problems I had while playing Team Fortress 2 was that I felt limited by the control scheme that Valve had set for me to play with. After much tweaking, these are the controls I have settled on.
The reason I like this way of playing so much is that you never have to take your thumbs off of the movement/aiming sticks while performing actions that require precise movement/aiming.
For example, you don't need to make aiming adjustments while reloading, switching weapons, or taunting. At the same time, it is easy to dodge and attack simultaneously without any trouble.
Setting the controls this way also makes crouch-jumping incredibly easy and Rocket/Sticky Jumping possible. To further increase the simplicity, I would suggest switching the crouch from Toggle to Hold.
Finally, the most critical thing you can do to your controls to improve your game is to adjust your sensitivity so that it is as high as possible. It may seem challenging at first, but in the end, it allows you to defend yourself much more quickly, and to attack both accurately and with a better response time.
And for those of you who really want to take it to the next level in any FPS title, I would highly recommend you invest $10 in these Kontrol Freeks. They may look bizarre, but they make sense scientifically. The greater leverage supplied by the extensions allows for minuscule aiming adjustments that are otherwise impossible.
I can testify that they have improved my skill significantly, and they are quite sturdy, so don't worry about them getting broken. Good gaming!