This started as a general weapons list, but I decided to instead focus on the classes giving you problems.
Having said that, one brief note before we get into class specifics:
- Airblast and Reserve Shooter is the "cheap" option because of how OP the Reserve Shooter is for the Pyro.
Dealing with Heavies
Heavy is fat and really slow. Because of this, the standard Flare Gun is your best weapon for dealing with Heavy. Be aware that it only does 30 damage to enemies that aren't on fire, but hitting them a second time does an instant 90 damage.
Flame + Airblast + Axtinguisher is another option, but the Axtinguisher got nerfed a while back, so now it only does mini-crit damage unless you hit them from behind.
Other solutions are... situational, such as the Manmelter's stored crits and the Phlogistinator's activatable crits. Do NOT activate the Phlog's right-click ability in front of a Heavy as he will kill you before the taunt ends even with a 75% damage reduction.
Dealing with Pyros
Due to the physics of Flamethrowers, if both of you are using flamethrowers, backing away from them as they move towards you causes them to take more damage.
The shotgun is really good against enemy Pyros, as are melee weapons (although that's less useful on nocrit servers). Airblast is also your friend as its good for keeping enemy Pyros away from you if you're using a secondary weapon to damage them.
Do. Not. Airblast. a Pyro who is doing the Phlogistinator taunt. It frees them immediately to start firing and gives them a longer Phlog crit thing.
Your flamethrower can make a difference too... the Degreaser is the worst flamethrower for a Pyro v Pyro fight unless you're pairing it with a secondary. The stock Flamethrower does more damage in a direct fight.
Dealing with Soldier
Learn to anticipate when they're going to fire rockets and airblast those rockets back in their face. Reflected rockets do mini-crit damage (to both them and you, so don't get too close).
Do note that rockets from the Direct Hit and Liberty Launcher are harder to reflect as they move faster than standard rockets.
Phlogistinator and Backburner are a serious liability when dealing with Soldiers head on as you have 0 and 4 reflects at most respectively. The other flamethrowers are all pretty good, though.
Smart soldiers will either try to shoot rockets at your feet (which are harder to reflect back at them) or simply pull out their secondary weapon. Unfortunately, the latter make it harder to deal with as you need to resort to your secondary or close the range between you.