There are 8 gym leaders, you have to specify which Gym leader you're talking about. If you beat the 1st one, you have to travel east until you reach a cave called MT. Moon.
Vermillion city (after beating the gym leader)
by beating the gym leader
try beating the gym leader
after beating a certain gym leader after beating the 4th gym leader aka vermillian
The 7th gym leader in Pokemon yellow is in Cinnebar Island, which is south of Pallet Town. You need a Pokemon with Surf to get there.
By beating the gym leader
Sabrina, the psychic gym leader. She is located in Saffron City just after beating Koga, the fifth gym leader.
after beating gym leader blaine
by beating the fourth gym leader
Brock is the first gym leader in pewter city.
Giovanni
The Only Way To Get Into The Gym In Viridian City Is You Have To Beat All 7 Of The Other Gym Leaders Along With Beating The Leader Of Team Rocket.....As Long As You Have Strong, High Level Pokemon You Can Beat Him...Hint The Leader Of This Gym Is Also The Leader Of Team Rocket Giovanni.....