Poliwag and Poliwhirl can be found in all three games in the following locations with the Fishing Rod and/or Surfing: Route 6, 22, 28, 30, 31, 35, 44, Violet City, Ecruteak City, Blackthorn City, Ruins of Alph - Outside, Ilex Forest, Mt. Silver - Outside, Viridian City
Poliwag can be found in Crystal during the nighttime in the grass: Route 22, 30, 31, 44
Poliwhirl can be found in Crystal during the nighttime in the grass: Route 28, 44, Mt. Silver - Outside
In Crystal, the earliest Poliwag can be found are Routes 30 and 31. In Gold and Silver, the earliest it can be found is in Ecruteak City.
A King's Rock can be found in the deeper parts of Slowpoke Well. It can also be found on wild Poliwhirl, Slowpoke, and Slowbro as an 8% chance.
From this point onwards, you can fight the gyms in any order, though you will need to retrieve the Machine Parts from the Cerulean City gym before you have access to the earlier portion of Kanto. Feel free to anticipate or postpone any battles as needed.
Gym #9 - Brock (Pewter City, Rock-Type):
Rival (Mt. Moon, optional):
Gym #10 - Misty (Cerulean City, Water-Type):
Gym #11 - Lt. Surge (Vermilion City, Electric-Type):