My Pyro strategy: Flamethrower, Shotgun, Axtinguisher (sometimes Fire Axe if I am feeling risky and dangerous)
-Ignite, blast, Axtinguish. Classic combo that works wonders for those who can use it.
-Ignite, blast, blast, Shotgun. Piss them off with juggles while they burn, then frag their airborn pinatas for 2 or 3 easy shots.
-Ignite, blast, Shotgun cornering. Works best when keeping distance from alert Heavies, retreating, or holding off an army of enemies.
-Blast, Shotgun, repeat. What I find to be the best way to deal with rival W+M1 Pyros. Works well with Fire Axe too.
Backburner is a lot more situational to me than the airblast, so I rarely, though not never, prefer it. Flare Gun can be very useful on defense, although I prefer the more direct killing power of the Shotgun, especially against Scouts, Heavies, and Pyros.