The jump key is seriously your friend. You want to learn how to travel using all of the classes as efficiently as possible. Half the fight is GETTING to the enemy.
Dont use the heavy without a medic nearby unless your enemies are complete noobs, you've got zero dodging capability even though your gun can rape-kill like nobody's business. Heavies aren't all that noob friendly like some would believe.
Dont be afraid to use melee, in the right circumstances it can save you ammo.
Don't be afraid to tackle the problems no one else will. Be the hero. If there's an imbalance on your team, don't wait for someone else to fix it. For example if there are no medics, be a medic.
Scouts can kill Heavies, just flank them to no end and they die very shortly. However you'll probably be putting all of your shotgun rounds into him when he drops dead ~ run, hide, reload, find some meds, then get back in the fight.
If you're low on health, retreat. This may seem like one of those unthinking run-and-guns but the same tactics that would help you in a game like Modern Warfare will help you here too. Use cover, wait for your health to get back up, (in this case, find medkits), never go it alone if you can help it. Dont follow everyone down the same corridor.
That being said, my best class is the scout. I love finding new ways into bases that really, only the scout, soldier, or demo man can get to. Thing about the Scout though is in tight spaces with sharp corners he can be out of a firing line in the blink of an eye while the other two have to absorb bullets. And with the addition of the double-barrel, there is no where you cant go.
Snipers, Medics, and medicless heavies, these are your targets when you play a scout.
Oh yeah, I wouldn't recommend engineer very early on. Not for a need of serious skills before playing the Engineer, but a bad Engineer can be kind of useless and attract a lot of hate.
If you're interested in playing an Engineer, watch the other Engineers, watch where they put their equipment down. When you feel like you can contribute with that class, always, ALWAYS build the ammo station (forget what it's called) first. It gives you metal to make more devices. Place that one down, then when you've got the metal, put down a simple sentry, then start upgrading the ammo station. When it hits level 2, upgrade your gun. When your gun hits level 2, upgrade your ammo station to level 3, then your sentry gun to level 3. Easy peasy. Dont build your sentry right next to the station, spies tend to go for the gun before they go to the station. If you're hanging around and someone is running right for your gun when the recharge station is 2 yards in the other direction ~ shoot them.
The spy is probably the most difficult class to master. Those disguise kits take a fair bit of forethought to use. Dont pick the scout ever, people can tell you're a spy because your running speed is no where NEAR that of a scout's. It's also a bad choice on maps with a very low number of players. Good players pay attention to the names, and could have met whoever you're copying less than a few seconds ago. When sabotaging engineer devices, go for the ammo station first.