Pyro is a class that is meant for hit-and-run tactics not "kill them all with fire".
They are quite fast and are good in assasinating and disrupting tough classes like soldiers, demomen and scouts. Pyro's mostly go alone not in groups due to that they are not good in group fights. You can be easily killed by snipers while in the open so think like a spy.

Tips for Pyro's everywhere!
-try not to W+M1 (if you do, at least pull out your shotgun once -.-)
-try to learn the degreaser and it's methods of use
-you are a very squishy class that can be easily killed when you are not noticing anything
-your flare gun crits when someone is on fire!
-Stick to enclosed spaces and corridors and ambush the enemy!
-Airblast... you know there is a M2 right?
-Time your airblasts
-practice your "burn-crit-ded" combo, especially on scouts/demomen

Smart plays for pyro: Try to hide in spots that are hard to notice. Once no-one has noticed you, surprise muthaf***as! That is what i meant with thinking like a spy. You may not be disguised or cloaked but it is the environment that will help you. You are crap in water so don't even try to go in it(if you have your shotgun up then it's ok).
This video might give some impression on what a real pyro is!

First and foremost: you are meant to be in the backlines, taking out key targets who are flanking your team. You are most effective when sniping on open ground like areas in Badwater, Hoodoo and many large maps. You needpatience and good reaction time for each kill. Once a shot has been fired, people will generally try to avoid you except for other new players. You are very effective in killing most classes from afar but you are also very deadly in close combat with the Jarate-Bushwacka combo (jarate someone and then kill them with a crit from the bushwacka). Targets like spies and scouts are hard to kill since that the spy can just deadringer through the open field and the scout is just too fast and can jump into hard-to-reach places.

Tips:
-Don't shake your mouse all to often, it will ruin your aim
-do not stay in "tunnel vision". pick your target,zoom,fire,ded not big surprise
-stay alert for spies since you'll stay in the same place
-try not to stay in the same place everytime you respawn, spies will recognize it
-use your jarate to bring mini-crits to the battlefield.
-your melee weapons are very effective at times

Smart Plays for Sniper: Jump if you can't see your target. Jumping cancels your zoom toggle and will help you when there are enemies nearby. As the Tau say in each battle: "the patient hunter, gets the prey", it means that you have to just wait for someone to get into your range of fire. Do not find try to find the target, but let the target go into your vision. You will deliver the killing blow as soon as the prey gets into your sights. When you see a Heavy going into your path-of-fire, kill him, for he is a deadly asset to the other team, providing supperesive fire to the battle

As the Spy you deliver the "Mont'ka" or the "killing blow". You have to infiltrate through enemy lines, and eliminate your target. You can kill any class with your backstab or kill the from afar with your Revolver. You are an effective support class since you can stab an entire front line within a few seconds. You are the rival of the Sniper, another assassinating class who delivers his killing blow form afar. Classes like heavies and soldiers are prime targets too but so are other classes.


Tips:
-STAY AWAY FROM PYROS
-Try to be like another team member on the other team while you are disguised
it will fool them.
-If you are caught stabbing someone alone, whip out the deadringer so you can feel safe
-Try to act natural while you approach your target
-Trickstabs are only used in a dire time.
-Pyros who have mastered the art of killing spies are deadly
-Normal cloaking devices are OK to use, but stay alert of your surroundings, you do not want to get shot.
-if the other team doesn't notice you, then try to eliminate your target as fast as you can
-Try disguise as Pyro, demoman, engineers and Snipers. They all have the same movement speed

Smart Plays as Spy!
Your backstab animation only comes once your near the back of the opponent so be wary about that. Blend in and try to act natural. Once a pyro is after your, shoot him/her as much as you can until backup arrives. They perform terribly in far range so try to headshot with your ambassador if you have good aim. Your deadringer should be on once you are passing through a crowded area because there are going to be filled with rockets.
If you are in flames, run away and hide. You also want to have your deadringer not activated once you are in the area of a Pyro. Flip it out once you are in flames so that the deadringer negates the flame.

I'm not the best in trickstabbing so you'll have to search for info about Trickstabbing through the internet for that, sorry TT