As a Heavy, who should I prioritize my fire at?

Based on this question.

I'm aware that there are certian priorities in taking down particular classes in TF2. Medics obviously are to be taken out first. Engineers and their buildings are a high priority as well. Any high-damage class that can be taken out is great.

But, as a slow-moving, close-range, big buff tank, prioritizing the guys behind the lines and the guys jumping over my head seems like not a terribly good way to make use of my ammo.

So I'm wondering how I should prioritize my precious target-space when preparing to shoot at enemies.

At what range do different classes become a viable target? Who should I try to focus down first? Should I completely ignore scouts? When should I switch from my shotgun (if I have one) to my minigun to maximize effectiveness against different classes?

In short, what class should I prioritize on my "Kill list"?

Take out the biggest threats who are in your optimum range. While the minigun can be used at very long ranges, it isn't terribly effective. So go for the biggest threat in medium-to-short range.

Medics, demomen, charging pyros, soldiers, enemy heavies. Roughly that order, as the situation calls for. Heavies aren't the best for taking out engineer nests, but they can do in a pinch if you don't have soldiers or demos doing it (and of course you should charge the nest during an uber).

Snipers will usually be too far away (and possibly with at least partial cover) for you to be effective. Scouts usually aren't the biggest threat, and they're going to be a harder target for you to hit, so they'd be lower priority. Engineers themselves shouldn't be a huge priority, but you probably won't see them other than when you're charging a nest as mentioned above. Minisentries are a joke to a heavy that you can clear out at your own leisure.

That just leaves spies -- you can do some spychecking in a pinch, and a spy getting in close range is an extremely dangerous target that should be brought down, but you shouldn't really be doing spychecking when there are other targets available (unless you notice somebody acting suspiciously) and hopefully you have a pyro or somebody else doing spychecking so you don't have to. Still, you may as well if you can spare ammo (or know you are about to refill)

A class priority list is not going to be useful because there are simply too many variables that change the priorities depending on the situation.

1. Shoot the closest target first

This is a decent general rule of thumb, regardless of what class the target is. At long distance, bullet spread and damage-falloff make heavies very ineffective. They are the kings of short / medium range. When someone on the enemy team runs straight into your team, it is mainly the heavy's job as a defensive class to kill him, for two reasons. First, the heavy's short range damage output exceeds even that of the demoman. Nobody can vaporize short range targets faster than a heavy if he is already spun up. The second reason is a minigun does not need to reload. If a soldier, demoman, or scout has to shoot the overextended enemy, they are using up loaded ammo to kill things the heavy could have killed faster anyways, ammo which they can use to shoot things farther away than you can shoot. Do not ignore the enemy running by because you think he is a "low-priority class" that someone else can kill while you are trying to shoot that fleeing medic from across the map instead. A corollary of this rule is you need to see someone in order to shoot him, so check in all directions around you and look all directions (including up) frequently. A good heavy should be a dizzy one.

Another part of this point is sticky bombs. Sticky bombs on the ground can be destroyed by bullets. Usually a demoman firing them at you / your team will detonate them immediately, but if he's slow, every one that you do manage to destroy is 60-120 fewer damage dealt to your entire team. However, sticky bombs are tiny and can be hard to hit with heavy bullet spread, so you should really prioritize players. However, in the absence of anything better to shoot, don't be the heavy that's wasting bullets on far away targets while there are undetonated stickies carpeting the floor right at your feet.

2. Shoot the greatest threat to your medic.

If you have a medic nearby, your top priority targets are the biggest threats to your medic (threats to yourself still come first - you are no good to your medic when dead). That scout that runs by, ignoring you, is probably trying to drop your medic. If he's any good at aiming it will only take him 0.625 seconds to fire 2 shots that do 80-105 damage each. Don't ignore scouts. Shoot them instead, and don't lose your medic to random scouts. Scouts often come in from flank paths or weird places so again, look around you often. If you are relying on your medic to ask you to turn around, it may be too late. A similar thing applies to soldiers. The only thing better than you at stopping a jumping soldier is a sentry, but either way you should make sure that soldier is pinned to the skybox with bullets.

3. Don't spend more than 3 seconds shooting at the same target at a time

Your close range damage output is insane. If you can't kill a target and they can't kill you in 3-4 seconds of continuous shooting, that's probably not a battle worth fighting. After 3-4 seconds, it's time to check behind you for any enemies that tried to make a move while they thought you were distracted. 3 seconds is about enough time for a spy to realize you are distracted and get you, so if you haven't looked all around you in the last 3 seconds, look again. Don't get fixated on mini-sentries or other heavies all the way across the map - fewer than half of your bullets are hitting the target at that range so don't waste your time / ammo.

Parting thoughts

In general a heavy's job is to provide a stronghold. You are too slow to spearhead a push as effectively as a soldier / demo can, but what you can do is spin up just behind the front lines of your team and give them a safe zone to fall back to, where any enemy that tries to invade will be greeted by a face full of your bullets (e.g. the above case of scouts / soldiers going for your medic). Remember that heavy is a defensive class, not a fragging class.