I have a flamethrower, which deals fire damage, but I don't know where to find the rules about it. Where in the core rule book (or any other book) can I find it?
Equipment rules for this weapon can be found on page 128 of the Star Wars: Saga Edition Core Rulebook .
The rules are written in the equipment description referenced above.
The flamethrower is a ranged weapon that shoots a cone of burning chemicals 6 squares long and 6 squares wide at the terminus. Make a single attack roll and compare it to the Reflex Defense of every target within this area. The attack deals 3d6 points of fire damage to the target; if the attack misses, the target takes half damage instead. A target with the Evasion talent (see page 50) takes half damage from a successful attack and no damage if the attack misses.
Fire damage behaves like any other regular source of damage (energy, sonic, cold, electrical), and these names for sources of damage can be considered as "flavor text", unless some ability specifically references a special interaction with a type of damage source.
The rules for fire damage are located in the Star Wars Saga Edition Core Rulebook on page 255. It states that a character or creature that takes fire damage is set on fire until they take a full round action to put the fire out. While a character is on fire, they are subject to a 1d20+5 vs FORT attack every round that deals 1d6 damage or half damage on a miss.