Is there any advice anyone can give me about keeping track of your character's position in team fights? Usually I end up lost in the middle, and end up just targeting someone and praying. Do I need to learn to play with a locked screen? (I really hate playing like that)
Any help would be much appreciated.

You can press/hold down the space bar to temporarily center/lock the screen on your champion. Other than that... some of it is just knowing in general where you are in the cluster, though when the enemy has a lot of displacement abilities it becomes more difficult.

If you have trouble keeping sight of your champion you could go for a ranged champ like Xerath who keeps a far distance when attacking. Holding down spacebar like Panzerfaust said tempoarily locks your screen to your champion which could help.

It takes practice. I still occassionally have trouble keeping track, especially if I'm playing an unfamiliar champion. Sometimes, if I really need to 'reset' my focus, I'll hit spacebar to center the camera and click just outside the heaviest fighting.

Comes with practice, I'm afraid.