1. Hello hello first things first: Yes, TF2, love it. Love the game, and I could give you plenty of tips, but before that, you should definitely give the TF2 coaching system a shot. It's hit or miss, but when you get a good coach, they're great. 2. And in answers to your questions:
Do I play? Yes, because it makes me happy (+ more reasons than I can get into now) There's a three-way tie for favorite but the honor of least favorite goes to Mavis Beacon Teaches Typing. What to recommend depends on what YOU like, so I'll present you the same question. Multiplayer or campaign? First person, third, or bird's eye? Etc etc etc...
What exactly IS the coaching system? And multiplayer, definitely. And fps etc doesn't really concern me. Anything with a nice multiplayer is good for me.
Then focus on TF2 for now. Find your favorite class(es) and go from there. As for coaching, it's essentially letting someone else ride around in your head while you play and giving you tips. The official wiki does a pretty good job of explaining it.