Where does a new Team Fortress 2 player start?

The point of the game is to work together as a team to defeat the enemy team. There are now two primary modes:

• Multiplayer - You and your team take on another team of human players in a game on one of several types of maps including:
  • Capture the Flag
  • King of the Hill
  • Payload (one team attempts to "push" a cart along a track to the other player's base)
  • Control Point
  • And others ...
• Co-Op (AKA: Mann vs. Machine) - You and your team work together to fend off wave after wave of AI controlled "robots".

Keep in mind TF2 is a "class based" shooter, which has nine roles, which each play very differently.

• Scout: A fast light class that captures points more quickly.
• Soldier: A slightly heavy class that carries a rocket launcher.
• Pyro: A short range, mid weight class with a flamethrower, which is great for rooting out spies.
• Heavy: A very heavy, slow class with lots of HP and a fully auto "mini-gun".
• Demoman: A mid weight class that carries explosive weapons, including the remotely detonated sticky bombs.
• Engineer: A mid weight class that focuses on building buildings on the map such as sentry guns and dispensers that supply health and ammo to your team.
• Medic: The ubiquitous healer with a heart of stone.
• Sniper: A mid weight class with a sniper rifle.
• Spy: A spy that can turn invisible, disguise as a member of the opposing team, and kill with a single backstab.

Your choice of starting class will likely depend on your experience in other games. Soldier, Heavy, and Sniper will probably be the most familiar to players of FPS. If you haven't played a lot of FPS I'd suggest Pyro, since it's a little less particular about aim, and doesn't require the strategy of Medic or Engineer. The game does feature training which I would suggest using before jumping in, it'll get you familiar with the classes and controls. After that find a multiplayer game and jump in.

You may also want to check out the Official TF2 Wiki.

It's largely a standard, team-based FPS multiplayer game, but only with the "strategic" game modes left in. If you've played King of the Hill/Capture the Flag most game modes are like that. There's always certain objectives instead of straight-up deathmatch, though Arena is close to Deathmatch, it's more like Counterstrike where you don't respawn, you just win when the other team is eliminated.

A good way to start is to pick a familiar game mode (hard to mess up Capture the Point or Capture the Flag) and pick an easy to use class. An attack class like Soldier or Pyro is easy to play in almost any mode as they require relatively little strategic planning, knowledge of the map or even accuracy (depending on your weapon). If you follow your team mates and shoot at bad guys with rockets you can be a bit of a help to your teammates and observe how other players and classes work.

Watch other players to pick up the game (or, if you have a headset, talk to them). You'll quickly pick up on where Snipers stand watch, where Engineers build their buildings, the routes Scouts take to grab the intelligence (flag), etc. As long as you play a class that can hold their own in the fray (pyro, soldier, heavy) you can pick up the game largely at your own pace without dragging your team down too much.

This isn't to saw Soldier and Pyro don't require any thought at all, but a brainless Soldier or Pyro is much more likely to be able to actually kill someone than a brainless Spy or Medic. These classes also let you follow around your squad as they attack/defend and you don't really need to do much aside from shoot at anyone who has a different uniform color than you. Also spies can cloak as members from your team, and since there's no team-damage, you might just want to shoot people from your OWN team if they're suspicious. Or if you're pyro just shoot everyone forever.

To learn more about any specific class or map, be sure to check out the TF2 Wiki. I'd recommend just jumping in the game with a simple minded class and consulting the wiki when you know what you want to learn more about though. The game is deceptively complex at a high level but simple enough to be enjoyable if you learn as you go as well.

Whilst the other answers deal with specific classes to choose and strategies to take, I'll start with a few in-game options and things that I believe bring new players quick benefits in game.

Disclaimer: I started with Pyro and Medic, but find that I use Engineer and Spy more often these days.

Options that should be turned on:

• Auto-Reload.
  It's interruptible (you can start firing again at any point) and it makes a ton of difference if you don't play FPSs often enough. (My itchy reload finger still jerks to 'R' out of muscle memory from other shooters, but it's good to have on just in case)
  Note that: As of the 2015 Gunmettle update, this option is on by default
• Hear a positive feedback sound when you hit an enemy.
  • Because everybody likes positive reinforcement.
  • Also satisfying as the Pyro when you flame someone and die, you still hear the 'ding's of them burning.
  • Can help give you a good sense of the range of a weapon - a Heavy's minigun for example has a bit of spread
• Players automatically call out when low health for Medic.
  • Gives you a handy little icon above players heads telling you who needs healing
• One click heal (As opposed to click-and-hold to heal).
  • If you're holding your Medigun (9 times out of 10 you should be) you should always try to be healing someone.
  • You build up an 'Ubercharge' meter whilst healing which (when full and upon activating) can turn you and the person you are healing invulnerable for a few seconds
  • You can 'overheal' teammates to (I think) 150% of their maximum health. If you stop healing them this slowly drops back down to 100%
• Quick weapon swap Scrolling the mouse wheel usually brings up a menu of your weapons, which you then click on to equip. Enabling this means scrolling will cycle through your weapons instead of bringing up this menu.
  • Less confusing, especially in the heat of the moment.

The next points have positives and drawbacks, I'd say try both sides out and pick based on your preference.

• Shorthand Disguise Selection for Spies. Replaces the numbers 0-9 for selecting a disguise with two groups of 3 (keys 1-3).
  • Positive Don't have to move your hand far from its primary position on the keyboard (less time spent re-adjusting to the keys).
  • Negative Have to press 3 keys instead of 2 ('Disguise As'->Class vs 'Disguise As'->Group->Class)
• Adjust Medic Auto-call Sensitivity again this is a personal preference.
  • I have mine set to 40% damage (60% health). Some people prefer lower, some higher. I tend to change healing targets really quickly unless deploying an Ubercharge (for the benefits of overheal, people on fire etc) so it helps me to know who's damaged in general.

These are just some of the options you can enable/disable, but they're the ones I've found most useful. I hope this helps.