Pallet Town doesn't have a Gym Leader. The town is just to peaveful and filled with memories, it doesn't need a Gym Leader. Although you can find Blue/Green's little sister. As well as I think Red's mom.
You need to go to the Burned Tower in the North West part of town. You need to go downstairs, the legendary dogs will then run away. The gym leader should be back at the gym.
go to pallet town, i think
lugia appears in the whirl islands.
Blackthorn city, (east of mahogany town).
Azelea town
in mohgaony town
Viridian city
It is in Mahogany town in the southwestern corner of the town.
beat the 6th gym leader and then go to mahogany town.
Mahogany town, just south of lake of rage.
Blackthorn City, after Mahogany Town and after Ice Path
Cinawood city, which is in the next town after the one with the lighthouse.
the next city with a gym is Azelea town it has bugzy as the leader
After beating Blue you go to Professor Oak's lab in Pallet Town.
you have to go to azalea town and beat the second gym leader, Bugsy.
blackhorn city i think its next to mahogany town