Unlike the original 1st Generation Pokemon Games, Pokemon FireRed actually has two different Pokemon Daycares. The first Pokemon Daycare can be found on Route 5, much like it was back in the original 1st Generation Pokemon Games. Unfortunately, this Pokemon Daycare can only watch one Pokemon at a time, meaning you can't breed Pokemon in order to obtain Pokemon Eggs.
The second and brand new Pokemon Daycare of the Kanto Region can be found on Four Islands after defeating the Main Story and obtaining the National Pokedex. Unlike the Pokemon Daycare on Route 5, this Pokemon Daycare can watch two Pokemon at a time. This successfully gives players the option to breed for Pokemon Eggs should they place two compatible Pokemon of similar Egg Groups within the Pokemon Daycare at the same time. If done properly, you will receive a Pokemon Egg from the Daycare Worker outside the building after leaving the two Pokemon in the Daycare for a while.