Try and stick to the Payload (pl_) maps if you can. In the general case the other game modes tend to suck in public games because multiple objectives is too much for the public to handle
I am only half joking.
I started by playing medic and I'd still recommend that today (5 years on). It's easy to start by healing someone (the easy option is the top scorer on your team who is a heavy/soldier/demo/pyro) and all you really have to do is try to stay alive and when your uber bar gets to 100%, right click when you think you or your heal target are in danger as it gives you both invincibility for 8 seconds. Following others around in this way gives you time to observe how experienced players play whilst still contributing to the team.
If you prefer to shoot people playing Heavy on a pl_ map is a surefire way to MVP. The cart gives you health AND ammo so you can just follow it and pew pew everything in your way.
The biggest tip I can give you to have more fun is to STAY ALIVE, especially when you are defending. The game really is about team work. I'm not talking about fancy pants strategies but simple stuff like if you are caught in a 1v2 or 1v3 situation (where you are the 1), get the hell out of there; one of the key differentiators between experienced players and newbies is that pros RETREAT. TF2 isn't like CS in that you can't get a headshot and kill one guy before you die. In TF2 even if you are a Heavy or a Soldier and it's 1v2 you don't get to kill one guy -- they'll smash you with their higher combined DPS and you'll barely take 50% off one of them. So the game is about playing position & numbers, solo charging will almost always result in you dying a lot.
This applies to the Heavy cart poosh I mentioned above too -- ideally you want to be on the cart as your team is nearby to support you.
ETA: use a mic if you can. Most folks will help you out in game if you speak up.