Do you get to travel to different regions in Pokemon FireRed?

Not regions seen in other games, but you can visit some islands that are introduced in the FireRed and LeafGreen games.
No. However you are able to visit the Sevii Islands, a pair of seven islands introduced in FireRed & LeafGreen.
No you do not get to go to other regions in Pokemon FireRed. You will remain in the Kanto region for the majority of the game, but you will be able to go to a set of islands.