I was thinking of posting a full guide but here is a class by class breakdown of how I play with the degreaser:

scout

The scout is tough to kill as a pyro as he can run circles around you and you dont have the weapons at range to kill him. Try and light them on fire and air blast nto a corner or wall and then shotgun to death if they run away, finish with the degrease if they get stuck or run at you again

demoman

airblast pipes away or dodge. Stickies can also be airblasted but dont run into a trap. Demos are pretty easy to kill if you have the jump on them, just do the axstinguisher combo for a 1 hit kill. If they are piping then try and airblast them into the air to mess up their aim. Demoknights are even easier to kill. Just airblast the charge, light on fire and hit them once, most are too retarded to realize that you are a stronger melee class and will run right at you.

Soldier

Good soldiers are tough to kill, even if you can airblast well. If you get too close and miss an airblast, they can one hit you. This is the problem most pyros try to hard to burn soldiers to death which will get you killed. If you can get the jump on them then a good axtinguish is best, otherwise airblast their rockets and dodge until they run out and then shotgun to death.

heavy

stay away from heavies unless they are focused on your team. It usually will take 2 hits from the axstinguisher to down them but if they dont see you then strafe around them so they cant turn while you wack at them. If they are with a medic, I usually burn both and can down the heavy even while being healed if they dont see you. If the heavy sees you, just run away and get some shotgun shells in.

medic

medics are too fast to axtinguish sometimes so I usually just burn them and airblast them towards my team or away from their team. When they run, which they usually will, just shotgun them

snipers

easy, just flames

engineer

stay away from sentries unless you can come from an angle that it cant see. You can get a jump on the sentry and down it before it turns sometimes. As far as the engineer, they usually are not on the front lines but if they are, just shotgun them as they usually will not stay close

other pyros

newb pyros will just run at you spamming W+M1. These are the easiest to kill. You can start shotgunning them until they get close and then airblast them back, shell them again or back away and flame and you shouldnt have problems. Against good pyros, if they are degreasing it will just be a shotgun battle, if they have the flare gun you will win easy with the shotgun, if they are backburn airblast them as far away as you can as you are at a disadvantage. Basically just shotgun other pyros is your best bet.

spies

you should be puffing flames at everyone and everywhere, there is no harm in it. If they are dead ringer then keep them lit and airblast them into walls so they cannot get ammo