The basics of Heavy should be engrained into your skull long before you dare peeking into this thread, simply because if you don't even know how to use the right-click to prep your mini-gun, then you're going to not understand what I'm about to talk about.
First off, basic pointers for using Heavy, things you should know no matter what:
1) Jump-Revving is the best way to prep your mini-gun. Strafe-revving slows you down significantly, leaving you open to being countered by spies from behind and snipers from afar. Jump-revving lets you not only move, but start firing as soon as your fat Heavy feet touch the ground. Especially useful in close-combat situations that you accidentally run into and can't escape.
2) Make sure the Medic dances in front of you when ubering you, if you want to take out Sentry Guns that are Level 3. Otherwise you're going to be pushed back.
3) Slow and steady wins the race. Don't turn a blind eye to your environment. Pay attention to who is behind you, and who is moving where. If you can not recognize a spy, Heavy class is a waste for you. Paranoia is your friend.
4) Those who are at a height advantage have a distinct advantage. Always get up as high as you can.
5) Know the terrain. If you can not find an ammo crate or med pack you are going to die, you are going to click on empty and you are going to waste your team's time.
6) Know the enemy. This is best experienced by playing all the classes. If you know how fast a scout moves you can anticipate it's movements better. If you know the best ambush spots for a pyro it'll make them easy to anticipate, as well as give you knowledge of places to ambush from.
7) Kill the closest first. All heavy weapons do best at close range. If a medic loses his healing target he'll often try to run away, leaving him open to parting shots.
8) The Shotgun rules. Do not fear the shotgun. If you can not aim the shotgun, learn. Sooner or later you'll have to rely on it.
9) You can dodge. Just because you move slow doesn't mean you can't. Jump, crouch, leap, strafe, stay moving and you'll piss the enemy off. It is possible to confuse the enemy by revving the minigun, and then as they start to fire stop revving while strafing. The sudden loss of the minigun will cause you to have a massive boost in speed, which'll throw their aim off momentarily. Especially useful for dodging long distance rockets.
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Map specific:
~ Well ~ Difficult map for the heavy due to the wide open, long distance and easily snipeable terrain. Concentrate your movements on the water-ways, as staying underwater makes it harder to snipe. While revving you can not swim, so use the shotgun until you get the hang of mini-gunning while underwater. Revving while leaping up out of the water can let you ambush the opponent.
~ Hydro ~ Stick to the more enclosed, twisty areas. The more open it is the more likely you're going to die. Heavy does best in close-range, ambush situations.
~ Granary ~ Easily one of the best maps for Heavy, due to the two million ambush spots. Keep to the high ground, concentrate on ambushing and advancing slowly. If you can get a medic, the middle point is your playground, so long as you watch out for snipers.
~ 2Fort ~ I don't have a massive k/d ratio on 2Fort for no reason. The single best ambush map in the game(In my opinion). DO NOT GO ONTO THE BATTLEMENTS. Not ever! Also, watch the compass to see what staircase the enemy is using, then get ahead of them and mini-gun away. Sewers are your friend, just beware spies who love to go down there.
~ Gravelpit ~ If you're attacking you are going to die a lot. Snipers love to pick you off. Try to attack from behind, use the routes le