TF2 is free to play, no literally you can play the game get all the weapons and stuff without paying anything. A premium account though lets you wear hats, trade with other users, and do a few other things. Hats though are an ectasy you will not be able to live without and you will cave in and thow all your money and valuable items you stole from your family and friends at your screen for precious hats.
It's a very team orientated game, there are no FFA's, every game will require a team effort to win. The main gamemode is taking and holding control points, that's as close to TDM as you'll get
(besides arena but no one plays that) There's also attack and defend, escorting a bomb to the enemy base, and some CTF maps. There's no real point to tf2 besides teamwork, winning, hats and having some goofy fun somewhere between. It's not just run around shoot at some people, there's objectives you need to work with your team in order to complete.
I would seriously recommend buying the Orange Box
http://store.steampowered.com/sub/469/ you get the Half life 2 series, Portal, and a premium version of tf2, and of course they all run through steam, as they're developed by Valve who made Steam. All great games in their own right, if you don't pay for anything on Steam it wont let you add friends or talk to people, unless they contact you first.
You should watch some videos on youtubes of the gameplay, and some tutorials.
A good intro by a guy who makes funny noises at the beginning of his videos and commentaries, and sports a pair of chops.
http://www.youtube.com/watch?v=6-CKMdIETAA
Good frag compilation from competitive play.
http://www.youtube.com/watch?v=NWrZVybyiU4
The wiki is a good place for info.
http://wiki.teamfortress.com/wiki/Main_Page