Here's the order you need to beat them in:
1. Pewter City: Brock (Rock type)
2. Cerulean City: Misty (Water type)
3. Vermilion City: Lt. Surge (Electric type)
4. Celadon City: Erika (Grass type)
5. Fuchsia City: Koga (Poison type)
6. Saffron City: Sabrina (Psychic type)
7. Cinnabar Island: Blaine (Fire type)
8. Viridian City: Giovanni (Ground type)
Once you beat all eight gym leaders, you can challenge the Elite Four and then the Champion, to become the Kanto Champion!