Classes

6v6 is played with six players on each team.

The common class format is:

1 Medic
1 Demoman
2 Soldiers
2 Scouts

Primary Classes

Standard play involves a team of two Scouts, two Soldiers, one Demoman, and one Medic. The Demoman is usually the damage dealer and the most important class following the medic. The scouts play similar roles, but one is usually supposed to be more passive than the other. The soldiers play completely different roles, the Pocket Soldier protects the medic and spearheads pushes, while the Roaming Soldier helps the scouts and is usually the first player to initiate, or to create a window of opportunity for his team. The passive scout stays in the range of the medic while damaging himself to help the medic build up his uber. The active scout helps the Roamer by baiting the enemy players to him.

Offclasses

In some cases, a team swaps a class of the cookie cutter lineup for a utility class, in order to create an advantage for your team, to help in a defensive situation or even to break a stalemate. This is called offclassing. A scout or the roaming soldier generally switches, which allows classes such as Sniper, Spy, Engineer, Heavy, or Pyro to be utilized. In general, offclassing means trading off the mobility of the standard cookie cutter lineup for damage or positional advantage or even a suprise effect.

Snipers and Spies are often referred to as "pick" classes, as their different playstyles and ability to 1-hit kill any class are very valuable when attempting to gain an advantage. Pyro, Heavy and Engineer are often called "defense" classes as they trade the mobility of one of the scouts or soldier, for better area control and/or damage output.

Sniper is perhaps the most common offclass, and a scout is the most common offclasser.

Practicing

There are a few different places where you can play practice games of various game formats:

TF2 Center - A successor to TF2Lobby catering to all levels of players, allowing players to join casual games of 4v4, 6v6, 9v9 (Highlander), Ultiduo and BBall. Offers optional mumble integration with hosted mumble servers. Available across all geographies, but particularly Europe, USA and Australia.

ATF2.ORG - A way to access to different IRC channels on the Gamesurge network, like #tf2mix (for beginners), and #tf2.pug.na (for higher level players)

TF2Pickup.net - Web-based 6v6/9v9/Ultiduo and BBall pickup site, catering to players with previous competitive experience. Considered to offer a higher level of competitive gaming than the others. Offers hosted game and mumble servers. Currently only available to European players, but promises global availability soon.

TF2 Stadium - "The newest and sexiest open source website for TF2 lobbies." - TF2 Stadium Steam group description

TF2 Pug.me - This is a higher level TF2 pug website where you will see anywhere from Top open to Top invite players.
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I will add on to this as the time goes by~