As it turns out there is no real way to learn all the ins and outs of this game. So i took the liberty to impart some wisdom on to you. All this material was learned at the school of hard knocks and is subject to being incorrect or not complete. It is also semi-opinion based as to my success and failures.
Ah yes I'll start with the noble Flamethrower. The Flamethrower is by far my favorite class and if used properly is devastating. A good EC build would be 3 in Helium Igniters and 3 in burst fire. BURN! is useful if you have time and are not worried about kiting large amounts because it will leave you pretty vulnerable. Frenzy is extremely useful for kiting or making up for low dps. For fighting Dome B use burst fire and a gauss or smg or chaingun because frenzy will make those mutas easy to kill. Get inferno and use on everything ground oriented. Exception would be the Black Ops. Inferno used to kill them but now they will shoot it down so it is worthless on them. Once you max out the pyro tree put points into vengeance if there is a slight chance you will be kiting. Vengeance is a great counter for Infested Marines. Vengeance is also great for mutas, which as a firebat you must use the knife, D, to kill. For kiting the actual queen nothing beats an smg and a flamer. I cannot stress the full beastness of the smg on all small zombies without armor. A flamer for kiting is also advised. Arc Reactors are imperative for the firebat. These nifty devices allow the firebat to spam inferno like a boss so try to get at least 1, 2 is better. Also the shield regen is handy. As you read on i will always stress the importance of having an ocular implant equipped for general purpose. It may seem like a waste but it provides much needed sight especially for kiting the queen and stinger use.