If you are new and want to play sniper, remember these things:
1. You are squishy. VERY squishy. The SMG is a pretty awesome weapon, but you often won't survive long enough to use it in direct combat, so avoid direct combat.
2. You can't shoot through glass or some other semi-transparent walls/fences.
3. No-scoping does almost NO damage and (I think) cannot headshot. The longer you stay scoped in, the more a bar in your scope charges. That represents your damage. It resets when you look up. Use your SMG or replacements for near targets, or just RUN AWAY!
4. Headshots triple damage, but sometimes a fully-charged bodyshot will suffice. Get a feel for when this is the case.
5. The only weapons that can headshot are the spy's Ambassador (the light-coloured huge revolver) and sniper primary weapons EXCEPT the dart rifle.
6. Remember that spys and pyros are the ones most likely to go after you. Plan escape accordingly.
7. Except in a very few gamemodes (Arena, etc.) KILLS DON'T COUNT; DEATHS DON'T COUNT. "Playing the objective" is a lot more important than popping X number of heads.
8. Only some weapons can overpenetrate. I think this is limited to the Machina when fully charged and the Huntsman bow.
Also, for all classes, remember the cartoon physics!
1. Most weapons have damage fall-off over distance. For shotguns and ESPECIALLY scatterguns, this is done to a ludicrous extent. In a few cases this is REVERSED to make unlikely shots that much more hillarious (Crusader's crossbow damage/healing, Sandman ball stun amount, etc.)
2. Rocketjumping/stickyjumping is expected behaviour. The devs even balanced for it.
3. Pyro airblasts and other teammates' liquid grenades (but not darts) will extinguish burning friendlies, as will being healed.