It learns it as a arcanine.
Growlithe learns Flamethrower at Level 35
Flamethrower: Growlithe can also learn Flamethrower via the Move Tutor in Goldenrod City (Crystal only)
Unfortunately, Arcanine does not learn Flamethrower. Your must level your Growlithe up to lv 50 to learn Flamethrower and then evolve it to Arcanine in order for Arcanine to have the move.
Vulpix learns Flamethrower at level 35.
train a charizard, he'll learn it eventually
Growlithe can learn the move Flamethrower at Level 34. This doesn't change if you give it a Fire Stone and make it an Arcanine either. ~Lucario
moves like bite, flame wheel, flamethrower lvl 50, take down
Growlithe does not learn Extremespeed, only its evolved form, Arcanine, learns Extremespeed at level 49. Note that this is the same level that Growlithe learns flamethrower, which Arcanine can only learn through a TM, so if you want both Extremespeed and Flamethrower, evolve Growlithe at level 49. Source: serebii.net
Arcanine doesn't learn flamethrower in any Pokemon games, if you want to teach him, give him the TM flamethrower (TM35). However, if you waited till lv 50, a growlithe can indeed learn flamethrower on its own. Unfortunately, arcanine would not learn extremespeed as a result. Ah but it does, if you get it up to level 50 it will learn flamethrower then evolve it using a fire stone and when it has evolved it will learn extremespeed soz if answer is a bit late
Route 8, 35, 36 and 37, you can catch a growlithe. Level it up and it will learn ember then flame wheel and at 50 it learns flamethrower. Evolve it with your firestone right after growlithe learns flamethrower and then have arcanine learn extremespeed after it evolves.
Use a Fire Stone on Growlithe. In my personal opinion, if you don't want to spent the money to get the TM for flamethrower from the coin trade in place next to the casino, I would wait till level 49 when Growlithe learns Flamethrower, and the use the Fire Stone, and then when you have an Arcanine it will learn Extremespeed also.
Palkia can learn flamethrower with a TM otherwise NO