You cannot get a fire stone until after you beat the Elite Four. Once you go to Kanto there is a guy in Cerulean City (where Misty trains). Then you go north to the road ahead where you can go either left to the Cerulean Cave which is closed for the GSC versions, or you can go right to the place where Bill's home is. Bill is not there, as he is in Goldenrod City. However, his grandfather is there. If you bring a certain kind of Pokemon to him each time than he will give you a certain stone.
Lickitung - Everstone
Oddish - Leaf Stone
Staryu - Water Stone
Vulpix - Fire Stone
Pichu - Thunderstone
This is the order that I believe that he will ask you for the stones. You need to show him the Pokemon. You may know this already but to answer you question again you cannot receive a fire stone for you Growlithe until you beat the Elite Four.