Also, specific class tips
Scout: Always stay on the move. Always. The double jump can get you out of a rocket splash damage, a Pyro's airblast, and it fucks up the aim of any Sniper that is trying to shoot you. The pistol is more effective at middle/long range than the Scattergun, don't waste precious ammo on far away targets with the Scattergun.
Soldier: Rocket jump. Not only does it get you places faster, but if you have a Medic with you it helps charge the Übercharge. Also you get higher ground resulting an advantage over ground level people. Fully utilize the splash damage of your rockets to launch enemies up in the air/away from you/off a ledge. Also try to lead your shots, and aim at the ground.
Pyro: Ambush. Way too many Pyros try and kill people from across the map by running at them holding mouse1 and W. If the enemy is too far away use the shotgun, it's more powerful than you can think (and it has a kill taunt for unaware snipers). If you have the Axtinguisher yet, you can take down heavier classes with easy, provided that you sneak up on them (which you should be doing anyway) by flaming them and then axing them. Also flame every suspicious teammate, they may be a Spy, and try to help defend Engineer's Sentry guns form Spies and Rockets/Sticky bombs. And try to practice the Airblast
Demoman: Sticky bombs. They are great traps, a powerful mid range weapon, and you can sticky jump. His grenades are also powerful, and are able to (but very very very rarely) one hit kill 125hp classes. Try to aim ahead of your targets while using the Grenade launcher, it has travel time, and it arcs as well. Planting a few stickies in the Intel/Point is a good tactic on defense, and with 175hp you are bound to be able to survive most attacks, just be wary of self damage
Heavy: Make sure to stay out of too open areas, Snipers will get you. Always watch your back for Spies, and if you have a Medic with you, communicate with him for great results. Your primary weapon does 500 DPS at close range, so you are pretty well equipped for short range firefights. The shotgun is great when you don't have time to spin up, or are out of ammo. If you have the Sandvich, the shotgun is traded in for a portable healthpack. Make sure to get behind cover while eating, or you may get killed. When approaching a corner, jump around it and spin your gun in midair, it gives enemies less time to run/prepare themselfs
Engineer: Make sure to build Dispensers near the frontlines for wounded teammates, and get teleporters up as soon as you can. On defense, the Sentry will help greatly, but that doesn't mean you have to sit behind it all the time, go out and help your team fend off attackers. Watch your back for Spies, and if possible get a friendly Pyro to help you guard the Sentry. Offensive engineering, while rare, is quite effective. Set up a level 1 Sentry at the frontlines, then lead people to it where you shotgun them while the Sentry shoots them. Don't place Dispensers too close to your Sentry, experienced Spies can use the Dispenser as a shield from the Sentry.
Medic: Don't stick to one person, heal everyone up to 150% health, it builds uber faster and it it more helpful to your team. Make sure not to heal any enemy Spies, or worse, Über them. When the Über is ready, communicate with your patient, and don't pop it until the very last second, when you or your patient are almost dead. Other Medics are your primary objective, heling them is more important than anything else, and burning teammates are more important to heal than non-burning ones. Also, get the Übersaw as soon as you can, it helps withSpies calling for Medic
Sniper: Shoot people in the head
Spy: There is just too much to know here, go somewhere else, watch OMFGninja's Youtube tutorial or something