I also just bought TF2, so if anyone could fill me in on the basics (unless there's a "tutorial" of sorts?) that'd be dandy.
1) Pick heavy as your class
2) never stop shooting
3) press C then 3 (repeat as needed)
With this winning strategy you shall be unstoppable!
Unless there is, for example, a sniper, or a spy, or a soldier with kritz, or a heavy with a medic, or a demoman, or... :p
Seriously though, @Apple. There are three things that will make your life a shitload easier in this game.
First, learn to dodge.
This means always strafe, circle, don't move in straight lines unless you're SURE you're safe. If it's an open area (with snipers) crouch erratically. If there aren't any soldiers around, jump as well. A great way to learn this is play medic. While you are healing a patient, you can't do anything except dodge, and since everyone goes for the medic, you learn to dodge really really quickly.
Second is situational awareness.
Basically, you should always know what is going on around you. That sniper dot on the wall, what colour is it? Keep an eye on the little kill list in the corner; how are your team-mates dying? How can you prevent falling to the same? Did the air just shimmer over there? Spycheck just in case. If shit goes south, what's the easiest way out of there? Where is the health and ammo? Are you on your own, or do you have backup? All this stuff is vital to know. A lot of people think they pay attention to what goes on around them. They don't. The easiest way to find out who does pay attention to these things is to find someone who is table-topping a heavy on a map that isn't heavy friendly (say a ctf, attacking cp side, or to a lesser degree, defending payload side), because, more than any other class, a heavy lives and breaths by situational awareness. If the player doesn't pay attention, he's nothing more than a big squisy sack.
Finally, alternate routes.
Valve have done an excellent job with most of their level design. There are rarely chokepoints that are the only way through. There is almost always a back route, a sewer, a twisted little gully, whatever. Better yet, at any given time, only one of these will usually be contested. So take the other one. Everytime you can get somewhere where the enemy doesn't expect you or want you to be, you're gonna get a kill.
It's by no means a complete list, as things like ammo management etc are also important, but those are the big 3. Anyway, hope that helps