Pokémon players: How can I find the 5th gym to obtain my badge?

•I have four badges so far. I’ve been all around that area and can’t seem to find the 5th gym anywhere. I obviously don’t have HM Rock Climb so I can’t climb anything. But I can Fly. I don’t know where to go and I’ve been stuck here for days.

•Also, I am stuck in Valor Lakefront because the little fat guy in overalls won’t let me go foward to Sunnyshore City. What do I need to do in order to pass?