When you're the Heavy Weapons Guy (HWguy), get used to using our Assualt Cannon a lot. The Cannon can't fire while in the air or while moving, your feet have to be planted on the ground. When firing against moving targets, try to predict their movements and have them run into your bullet stream. It may also be a good idea to sweep the area, in order to catch those enemies low on health. Watch out for your teammates, though. You don't want to wear away all of their armor.
The major weakness of this class is its slow speed and lack of mobility. But you do have mad health and armor so don't be afraid of takin' some shots when charging a room. U can take about 5 good rockets before you die. Recent versions of TeamFortress have given the Hwguy a small break with movement. You used to have to completely stop firing in order to move. Now, if you try to move while shooting, the AC will go on "hold". It will make a distinct noise (the high pitched hum will be heard, not the full shooting sound) and you stop firing. You can then move around with the gun on reserve, but you will be moving at a slower speed (like the Sniper will when projecting his red dot).
The advantage here is that as soon as you stop moving the gun will fire back up at a quicker speed than it would if you had completely turned it off. Great for situations that require you just to strafe a bit or move around a near corner to get the enemy back in your line of fire.
Overall, the HWguy is best for defense. Remember you have the same MIRV grenades as a Demoman, but be careful because you only start with one. Let the enemies come into your line of fire by parking in a narrow hallway or a corner. Beware of enemy grenades bounced into your area though.

AGAINST

One of the weaknesses of the Hwguy is that the AC firing sound starts up before it actually starts to fire. So, listen for the sound. If you see a Hwguy firing, and he's not doing so at you, then blast away before he has a chance to stop! The Hwguy won't be moving much at all. So set up that Sniper Rifle, load up the rocket / grenade launcher, prepare the railgun, and have fun. Hell, you may even want to consider throwing a hand grenade or two at the dude, if you think he's going to be there for a while. Remember ... while the Hwguy himself can't move while firing, rocket blasts can still move him around. Don't fire two rockets directly at him, unless he's backed into a wall. Better yet, use the first rocket/grenade to knock him into the air. Try to predict his landing, and then fire again.
If he's firing at you, then running away should be your first goal (if you're still alive that is). Head towards a corner or spot where the Assault Cannon can't get to you, forcing the Hwguy to stop. This is your chance to come back out and retaliate. It'll take some time for the AC to rev up again, so be sure to attack quickly. If you think you really got skillz, try circle-strafing around the HWguy as he fires. When you do this, be sure to launch a grenade at his feet too, he'll be too busy trying to nail you to notice the phat gift you left him.