Here's my following tips for each class:
Scout:
NEVER use the Force-a-Nature. In most cases, the Scattergun will win over every weapon -- it does just around as much damage, but it's more reliable, carries more ammo, and doesn't have a stupid knockback.
Play this class as you would a Spy -- never go into the front lines, instead, sneak around and attack oblivious enemies with a point blank hit.
Soldier:
The Rocket Launcher's speed and splash damage takes getting used to. This is an ideal easy to learn, hard to master class. Primarily stick to the default Launcher, since most others use the Rocket Launcher's rocket speed and splash damage.
The Equalizer is unreliable, stick with the Shovel.
Pyro:
The Backburner is a bad choice. You sacrifice helping your team with the airblast to instead do more damage. Airblast is great when used right -- extinguish team mates, blow back Ubers, reflect projectiles, knock enemies off high heights or control points. With the Backburner, you practically can kiss that airblast goodbye.
Homewrecker should never be used for destroying buildings, only for destroying sappers, it's better to use your flamethrower.
Degreaser + Axetinguisher is the best combination.
Demoman:
This class is very similar to Soldier, except it's less beginner friendly. The grenades are harder to predict a point to hit someone, since they bounce, they arc, and they explode after several seconds. But, the grenades are powerful at every range, and a direct grenade hit does a lot of damage.
Use your sticky launcher, not only is it good at area denial but it can limit the flow of enemies.
Heavy:
Sasha is the better Minigun -- Tomislav doesn't fire fast enough, Brass Beast is way too slow, Natasha is too weak.
Always right-click your Sandvich to save time and your life. Remember to heal your Medic with the Sandvich if needed.
Engineer:
First, study a map and see where most Engineers put their sentry gun. That's usually the best place to put them, but for the love of god, do not build a sentry gun right next to another persons.
If your buildings suddenly are covered in stickies, an unstoppable Uber, or your Sentry isn't doing well, for the love of God, save yourself.
If your Sentry is ever sapped, follow this graph:
1. Do you know who the Spy is? Fuck your buildings, kill that Spy.
2. Are team mates around who saw what happened? Unsap your buildings, let them kill the Spy.
3. Did the Spy break disguise? Unsap your sentry, but don't give the Spy an easy backstab.
4. Did the Spy run away, or you don't know who it is? Unsap your buildings, be on high alert.
Medic:
The Quick Fix does not support pocketing (healing one team mate in particular), it supports healing an entire team.
Kritzkreig does best on Defense.
Never use the Bonesaw, always use The Solemn Vow or the Amputator at least.
Keep moving. A medic should be everyone's first target, so never give anyone a chance to take you out.
Sniper:
Keep moving. Keep moving. Don't stand still. Keep moving.
Kill an enemy (especially another Sniper), they'll know your location if you stand there. Move to another location at least every two or three kills. Never get in a Sniper war, that's just counterproductive to your team's efforts. Always take out the slower enemies (Heavies, Soldiers, Demomen). Never use the Razorback, rely on your hearing and erratic locations to keep Spies out.
Spy:
Very hard class for newbies. It's a class that takes high risk, high reward to a new level -- a good Spy can hold back an entire team, but a bad Spy will be nothing but a player who helps fuel the other teams' Ubers, Buff Banners and Revenge crits.
First of all, never ever ever use the Cloak and Dagger. The Invis Watch is better in every way -- longer cloak time, can be refilled with ammo, supports a fast and unpredictable Spy. The Dead Ringer is great for fooling an enemy, but a CnD spy is no help whatsoever, all you'll find yourself doing is sitting in a corner staring at opportunity walk by because you won't move.
Next, take risks, especially with Your Eternal Reward or if you're wielding the Dead Ringer. If anything goes wrong, cloak, run away, change disguise. If a backstab fails, don't keep stabbing your enemy in the face. Either cloak and run, or Revolver and backpedal.
General TF2 tips
Everyone is a Spy unless proven otherwise.
Never attack a Heavy from the front if they have their gun spun up.
Medic gets the Health Pack first, no exceptions.
Engineers get the teleporter first, no exceptions.
Aim for the Medic first.
Use voice commands (Incoming, Help, Thanks)
Kill the Engineer, THEN the sentry.