A basic guide about the flamethrower.
Finding the Flamethrower
The flamethrower can be found in weapon shops and you can pick it up from Reaping trucks or dead cultists wielding them. In weapon shops they are under the far right tab along with the bows.
At full ammo, the flamethrower comes with 300 ammo, with 100 being in your clip and 200 as spare, but it chews up this ammo fast. The flamethrower's ammo can be refilled like other weapons, buying ammo at the store or picking it up off of enemies.
Using the Flamethrower
I've found that to get the most out of the flamethrower you just need to give most enemies a single puff of flames to set them on fire. Listen for the screams of agony to confirm that you have set them on fire. While on fire they will run around screaming until they eventually die. Unlike most enemies, the heavy gunner and flamer cultists will take more than a single puff. Now go burn the cult down!