Basic Basics
Don't know what you consider "basic" but I'm going to assume you've got the
move and point and shoot parts down.
Maps
Capture Point. If a map starts with "cp_" then it is a "Capture Point" map.
Some of these go one way only, like Dustbowl. The blue team tries to capture, the red team tries to defend. Other ones, like Black Mesa, can be pushed back and forth. Capturing is just a matter of standing near the "point" which is usually where everyone is running towards. Follow people, do what they're doing, and you'll figure it out quick enough.
Capture the Flag. If a map starts with "ctf_" then it is a "Capture the Flag" map.
These maps are about snagging a briefcase or the "intel" from your enemies' side and taking it back to your team's side of the map. This is often a matter of trying hard to find a crack in their defenses. Explore your side of the map well before trying to infiltrate the enemy base, since these maps are usually the same exact layout on both sides.
Payload. If a map starts with "pl_" then it is a "Payload" map.
These maps are like Capture Point maps, but the "point" is a cart that moves along a track. If you're blue, stand by it and it moves forward. If you're red, stand by it and it stops. As a payload regular, I can tell you that if you want your team to love you, go stand by the cart. Everyone else tends to forget about that part. If you get lost, follow the "train tracks."
There are other map types, like arena and territory capture, but they don't see much light of the day on popular servers.
Character Classes
Pay attention to the notes the game shows you, they ARE the basics.
Scout. Never stop moving. Shoot people up close. Avoid pyros and heavies. Teammates will expect you to stay on the point/cart/intel since you're fast.
Soldier. Take care not to shoot too close to yourself. Aim at your enemies' feet. Harass enemy snipers. Stand out of sentry range and take them out. Shoot at your feet to "rocketjump."
Pyro. Burn enemy groups and run away. Stop ubered enemies with your compression blast. Teammates expect you to "spycheck" -- burn everyone and everything to see to find hidden spies. If you don't bother with it, teammates will be irritated.
Demoman. "Spam" your grenades -- just keep firing. People will expect you to take out engineer nests by standing out of the sentry's line of sight, firing many stickies close to it, then detonate. Don't blow yourself up. Avoid pyros.
Heavy. Stay near the point/cart on cp/pl maps. Defend chokepoints. Keep your gun spinning (right mouse). Avoid snipers. Watch your back for spies. Keep your aim just ahead of where your enemies are headed, they'll run into your bullets. Make friends with a medic, you'll be very hard to kill.
Engineer. On defense, always have a sentry up. On offense, always have a teleporter up. Watch where other engineers build stuff, they usually know the good spots. Build sentries to cover other sentries, it'll prevent the spy stab-sap maneuver. DO NOT build sentries next to each other. Help other engineers before building your own stuff.
Medic. Never, ever, stop healing people. Heal people who need it first, not just people who call for it. When you get uber (100% on your ubermeter, right-click to use it), don't sit around. Find a heavy or pyro to uber, or demoman when on offense. STAY CLOSE to them, they won't appreciate losing it before they can get their job done.
Spy. Spy is hard to play. Disguises always fool sentries and rarely fool anyone else. Stay cloaked (right-mouse) whenever you can. You can't stab or sap while cloaked. Gobble up ammo boxes like candy. Approach enemies from behind, but avoid pyros no matter what. Sap sentries on offense and teleporter entrances on defense.
Sniper. Aim for the head. Your teammates will expect you to keep them safe from other snipers. You can be very good at taking out sentries if you stand out of their range. Be a coward, it works.
