Here's some tips I'll share:
Spy: Pressing the minus key when the disguise menu is up allows you to switch colour (i.e. your own team), pretty useful to play mindgames with the enemy through the starting gates, and as someone else said it actually looks less suspicious going into enemy territory disguised as your team rather than one of theirs sneaking around. In sudden death, backstab the Engineer and their buildings are instantly destroyed with him so you aren't killed (unless there's another Engi's Sentrygun nearby). His Revolver isn't too shabby too. Oh, and another thing, don't sneak around just after you teleported, since your feet will still be glowing, let it fade first.
Engineer: Place an entrance teleport the second you're out of the spawn, whether your team is defending or attacking, then rush inside to fill up on metal before catching up with your teammates. Also, try and 'memorise' your teammates, i.e. what their playstyle is like, or at the very least what class they are, so you can spot spies a mile off before they even get close to your buildings.
Sniper: Charging up to max isn't always necessary, I think the maximum damage it does is 450, and the maximum health any class has is the Heavy at 300. Headshots just guarantee a crit (never seen a torso crit), so you could charge up very little and get in more shots in the same amount of time, good for target practice. Also stay zoomed out every 30 seconds or so if you're rather paranoid about spies sneaking up on you, your machete can make mincemeat of them pretty fast.
Scout: Double-jumping is your friend, takes some practice, but even a good soldier won't be able to get his rockets to follow a 180 degree leap, just stay away from Pyros at any cost and run to the safety of water (if there is any). The Scattergun is ok at mid-range but if you are better at evasion than aiming, just empty your rounds and run off to reload and repeat, or pursue someone retreating as more often than not they are on very low health which one or two Scattergun rounds will fix.
Heavy: Pretty hard to be any good without a Medic by your side, since the wind-up delay of the Minigun is a right pain (although you have a backup Shotgun) and can be strafed/kited by most classes due to your lack of speed and maneuvrability, although they are rather good for suppressing fire if there are lots of Demos/Soldiers/Pyros about pushing forward.
Demoman: In the right hands this is a deadly class, even great for offense. Great for mobile defending such as a dropped briefcase, or doorways (NEVER let them see even 1 sticky bomb visible for maximum effect, best examples on Gravelpit). Play around with the physics, to get a feel for where grenades usually land (note: every grenade bounces differently depending on how it's spinning), keep a backup of sticky bombs next to you for CQC where the grenade still has a chance of missing. Rig up engineer's buildings if there's a lot of pesty spies around, if it's congested it can sometimes be quite hard to pick out the spy for a direct shot, and blow the whole lot whenever you hear an Engineer's voiceover. Also great for your capture point when you're elsewhere spamming grenades.
Soldier: Old-school rocket launcher, but with a limited clip. Shotgun is handy for backup if you've used all your rockets and haven't had time to reload, Shovel for pure comical kills.
Pyro: These get eaten pretty easily if directly taking on a Soldier, best is to pounce on them from around a corner, since most Soldiers are pretty suicidal, especially when they are close to dying, usually taking you with them. Great for cleaning out Scout rushes on Granary, Flamethrower is useless in water. You are the second fastest class in the game after scout (I think), and with some skill you can even out-strafe a Heavy into oblivion if he's too slow turning. The ultimate anti-spy class, a good Pyro should NEVER be killed by one.
Other tips: In Sudden Death mode, or normal respawns, if you have turned off 'Suicide on changing class' in the multiplayer options, you can switch classes without dying, pretty useful in SD to switch to Medic, since usually the team with the most medics wins.
This is probably obvious, but the odd fizzles/clunks/whatever from your shots are guaranteed to crit, (blue Soldier's rockets are pretty easy to see in this respect, bright blue rocket trail!), and the Demoman's sticky bombs leave a sparkly trail. Another thing to note from the developer commentary, is that the critical rate is semi-random and that the better you play in a round and usually the lower your health, the higher your crit chance will be to promote those good runs that people love to brag about.
Spy: Pressing the minus key when the disguise menu is up allows you to switch colour (i.e. your own team), pretty useful to play mindgames with the enemy through the starting gates, and as someone else said it actually looks less suspicious going into enemy territory disguised as your team rather than one of theirs sneaking around. In sudden death, backstab the Engineer and their buildings are instantly destroyed with him so you aren't killed (unless there's another Engi's Sentrygun nearby). His Revolver isn't too shabby too. Oh, and another thing, don't sneak around just after you teleported, since your feet will still be glowing, let it fade first.
Engineer: Place an entrance teleport the second you're out of the spawn, whether your team is defending or attacking, then rush inside to fill up on metal before catching up with your teammates. Also, try and 'memorise' your teammates, i.e. what their playstyle is like, or at the very least what class they are, so you can spot spies a mile off before they even get close to your buildings.
Sniper: Charging up to max isn't always necessary, I think the maximum damage it does is 450, and the maximum health any class has is the Heavy at 300. Headshots just guarantee a crit (never seen a torso crit), so you could charge up very little and get in more shots in the same amount of time, good for target practice. Also stay zoomed out every 30 seconds or so if you're rather paranoid about spies sneaking up on you, your machete can make mincemeat of them pretty fast.
Scout: Double-jumping is your friend, takes some practice, but even a good soldier won't be able to get his rockets to follow a 180 degree leap, just stay away from Pyros at any cost and run to the safety of water (if there is any). The Scattergun is ok at mid-range but if you are better at evasion than aiming, just empty your rounds and run off to reload and repeat, or pursue someone retreating as more often than not they are on very low health which one or two Scattergun rounds will fix.
Heavy: Pretty hard to be any good without a Medic by your side, since the wind-up delay of the Minigun is a right pain (although you have a backup Shotgun) and can be strafed/kited by most classes due to your lack of speed and maneuvrability, although they are rather good for suppressing fire if there are lots of Demos/Soldiers/Pyros about pushing forward.
Demoman: In the right hands this is a deadly class, even great for offense. Great for mobile defending such as a dropped briefcase, or doorways (NEVER let them see even 1 sticky bomb visible for maximum effect, best examples on Gravelpit). Play around with the physics, to get a feel for where grenades usually land (note: every grenade bounces differently depending on how it's spinning), keep a backup of sticky bombs next to you for CQC where the grenade still has a chance of missing. Rig up engineer's buildings if there's a lot of pesty spies around, if it's congested it can sometimes be quite hard to pick out the spy for a direct shot, and blow the whole lot whenever you hear an Engineer's voiceover. Also great for your capture point when you're elsewhere spamming grenades.
Soldier: Old-school rocket launcher, but with a limited clip. Shotgun is handy for backup if you've used all your rockets and haven't had time to reload, Shovel for pure comical kills.
Pyro: These get eaten pretty easily if directly taking on a Soldier, best is to pounce on them from around a corner, since most Soldiers are pretty suicidal, especially when they are close to dying, usually taking you with them. Great for cleaning out Scout rushes on Granary, Flamethrower is useless in water. You are the second fastest class in the game after scout (I think), and with some skill you can even out-strafe a Heavy into oblivion if he's too slow turning. The ultimate anti-spy class, a good Pyro should NEVER be killed by one.
Other tips: In Sudden Death mode, or normal respawns, if you have turned off 'Suicide on changing class' in the multiplayer options, you can switch classes without dying, pretty useful in SD to switch to Medic, since usually the team with the most medics wins.
This is probably obvious, but the odd fizzles/clunks/whatever from your shots are guaranteed to crit, (blue Soldier's rockets are pretty easy to see in this respect, bright blue rocket trail!), and the Demoman's sticky bombs leave a sparkly trail. Another thing to note from the developer commentary, is that the critical rate is semi-random and that the better you play in a round and usually the lower your health, the higher your crit chance will be to promote those good runs that people love to brag about.