If you've been play Team Fortress since either TFC or Quake TF, you'd know that being killed by a pyro was on of the most humiliating things, especially in league play. Infinitely more humiliating than being killed by a scout's crowbar or engi's wrench. This class has finally found a niche after being the still-born bastard child of the game.
Let them eat napalm
Always try to lite as many people on fire as possible, the attrition damage last a very very long time. Lighting a large group on fire is a good way to stop an offensive push in its tracks or break up a defense, especially if there is no enemy medic on duty.
Be the ambush predator
You are not fast, despite being classified as 'offense', running straight at someone with your flame thrower on as they back-peddle is a real quick way to become another notch on that soldier's rocket launcher. Instead, hide somewhere you know people will pass and strike them from the side, at point blank range you can damage people A LOT faster than a medic can heal.
Strike and Retreat - DO NOT CHASE
As with playing ambush, you do not have the health to dance with the heavier classes, lite them on fire and retreat while pumping buckshot into them. Let the fire do the rest of the damage as you run away to heal.
Ax Kills = Biggest Humiliation
If you are close enough to use the Ax, you are close enough to use the flamethrower and do a ton more damage, but if you want to humiliate someone feel free to go at them like Jack Nicholas in a snowbound hotel.
Your Role is Not Offensive
Despite being classified as offense, the pyro is better classed as support. You are not as fast as the scout or do as much damage as the soldier to be effective on Offense, at least on CTF maps. Sometimes its better on D to sit and wait to ambush the enemy's offense, other times its better to go on offense depending on the enemies class mix.
Invisible spies are Everywhere and the Man is out to get you
If anyone looks suspicious test them by lighting them on fire, you are the teams first line of D against spies. When you hit them they light up like a christmas tree. If you think theres a cloaked spy around, hold down the fire button and wildly swing your torch around. Even if you don't think theres anyone around, just tapping the fire button every once in a while can uncover spies.
Despite wearing a gas tank and plastic, Other Pyros are not flammable
Only direct damage with the flame thrower will hurt other pyros, put the flamethrower away, take out the shotgun and back peddle. While they are still dancing in circles trying to hit you with the flame thrower you'll tear them to pieces with the shotgun.
Partner with a spy
A very good way of taking out sentry guns is to have a spy sap one and have a pyro rush it. The flame thrower does a lot of damage to engineer deployables, but odds are you will not get a chance to get close enough to them. Using this system, your flame thrower will actually go through the sentry model and burn the engineer and likely his dispenser as well.
Crowds like a good Bonfire
By far the thing the pyro is best at is clearing out a capture point on a map like Gravelpit, one or two pyros can clean out an entire team. Flip on the torn and run into the fray, people panic when this happens, but make sure to run out when your health is low and let the fire to the rest of the work for you.
Just remember, when playing pyro, you are not the useless piece of garbage you would have been in previous TF iterations, a good pyro makes life a living hell for spies and heavier classes. just remember, play like the name of the class, lite as much shit on fire as you can without becoming toast yourself.