Got bored, so I'm gonna make a small guide for MvM because I play it too much

Weapons
Phlog, Backburner, and Dragon's fury are fine, degreaser is not, and stock shouldn't really be used either. Gas Passer (or maybe scorch shot) should be used for secondary and the powerjack for melee

Primaries
Primaries depend on the situation.I usually go with the phlog but most flamethrowers are fine (not degreaser though, don't go degreaser.) Each flamethrower has it's differences. Phlog has guaranteed crits for 10 seconds on enough damage (but no airblast or random crits), backburner has guaranteed crits on flames hitting the back of an enemy (but airblasts cost a lot more than stock), Dragon's fury deals a lot of damage in a short time (but you need to aim), and stock is stock (but it's stock). Tanks can be shredded with the phlog and generally the phlog is the best choice if you do not intend to airblast ever. Dragon's fury is also a good dps, however you do need to aim more than the traditional flamethrower and enemy pyro's have a very low reaction time so they can airblast your shots which is a headache to deal with. Backburner does crit damage upon hitting the backs of enemies, but it requires you to flank and if that bot turns around you no longer deal extra damage.

Secondary
Gas passer is ridiculously overpowered in MvM once you get the explode when ignited upgrade. If you have a collection of many bots in one area just throw down the gas and ignite. It can instakill all normal robots except heavy, which will have little health left. And because it recharges on damage dealt you get the gas back instantly, allowing you to rinse and repeat. Scorch shot could also be used to push the bomb bot away from the hatch or into a death-pit to reset however that should only be used sparingly as it knocks back the target and without warning your team might miss a headshot or be confused as to where to shoot for a few seconds

Melee
Powerjack is the best melee for the situation. The increase in movement speed whilst equipped will allow you to get to the front lines easier, which is good if your team is missing a teleporter or you want to make a retreat for ammo. There's also 25 health on kill but you won't be using your melee much so it doesn't matter much.

Upgrades
As a pyro you are going to be up close to the robots to deal damage to them. So for first upgrades it should be health on kill for your flame thrower. It's also recommended to get some Über canteens for the first few waves as you may get overwhelmed without resistances. Next upgrade should be your gas passer explosion upgrade, then a healthy mix of flamethrower upgrades (if you're not a fire-and-forget kind of guy you shouldn't need to upgrade your burn time and burn damage) and resistance upgrades. My usual strategy (probably not the best strategy) focuses on out-damaging robots, so I forget most resistances and go with a phlog with max health on kill, some damage points & ammo, and Über canteens when the going gets rough.