After looking at every location in FRLG, it is obvious that Magikarp is the only thing that can be found with the Old Rod.
To get Goldeen, you need the Good Rod. The Good Rod can be acquired in Fuchsia City next to the Wardens house by the man with the lost teeth. These are the locations that Goldeen can be fished with the Good Rod.
Route 6, 22, 23, 25, Berry Forest, Cape Brink, Cerulean Cave, Four Island, Fuchsia City, Icefall Cave, Ruin Valley, Viridian City, Safari Zone