Here's just a compilation of tips that I thought would be helpful for even the pros.
1. Know how much hp all the classes have, this is very important for know if you'll have to charge a shot or not. If a pyro is charging at you, I know he has 175 hp max, so a single quick-scope headshot just won't do. So I retreat while giving him the occasional no-scope and my team can help finish him.
2. Take note of who enemy medics are healing. If a medic just overhealed a sniper, I know that that isn't going down in one shot, so either wait out the overheal, kill the medic, or kill the sniper with a charged shot.
3. Learn an enemy sniper's style. A sniping style is like a fingerprint, no two are alike. When I first join a server and enagage another sniper, I try to get the feel of his style, once I understand it, I can counter it and even identify him out of a mob of snipers. It's a highly useful skill.
4. Learn the subtle weaknesses of spies. These deadly dagger dudes are a sniper's worst nightmare, but there are two things that will give them away, and I'm not talking about a pyro 'on your team' running from the enemy base yelling "Medic!" Much more suble things. The first is that you can actually hear a spy uncloak, as well as his footsteps, just listen carefully and you might even surpise the spy with a krukri (my little name for a crit-kukri) to the face. The second giveaway is that when a spy come out of water onto land, you can see little water droplets around him, even if he's cloaked. I get so many accusations of aimbotting when I snipe in the 2Fort sewers.
5. Learn to do close-range headshots (Thanks, Koff, almost forgot about this). This ability has litmitless rewards, start by learning how to n-scope effectively, then take it a step further. But thats to a valve update, you can't get a headhsot withing 200 milliseconds of scoping in, making this a little more difficult. It also helps to get a crosshair for your sniper. Just go to multiplayer options and you can get one.
Anyway, hope this helps, even if I get some scrub to headshot a scout, I'll feel good inside... Unless the scout is me, then I'll switch to sniper, and I will find you.
EDIT: Removed spray section due to massive flames about it.