After you explore Fuchsia City and your partner pokémon learns some new Secret Techniques, you should challenge the Fuchsia City Gym.
Get into the Gym
You will only be allowed into the Fuchsia City Gym if you have caught at least 50 species of pokémon. If you haven't, use your Pokédex and press the + button to only display the pokémon that you have seen but not caught. Check the habitats of those pokémon, then try going to those places and catching them. Or just go to a place with grass, or surf in some water, and look for pokémon that you haven't caught before.
There are invisible walls in this gym that can only be seen when the mist swirls up from the floor.
First, you need to go east, where you can battle Tamer Phil, who has a level 41 Nidorina and a level 41 Nidorino. You get 2460 Pokédollars for winning.
Next, go northwest, then when you can't go farther north, go northeast to go through a gap. Here you can battle Ace Trainer Kayden, who has a level 41 Muk and a level 42 Beedrill. You get 4200 Pokédollars for winning.
From there, go northwest, then west. When you are near the corner, you can go south, then east to battle Juggler Kirk, who has a level 40 Haunter. You get 1600 Pokédollars for winning.
Then you can go south to battle Juggler Harry, who has three level 40 Haunters. You get 1600 Pokédollars for winning.
Go west from Harry, then go south when you can, then southwest, then go east, then south when you can. Here you can battle Tamer Edgar, who has three level 41 Arboks. You get 2460 Pokédollars for winning.
You can go northeast from there to battle Ace Trainer Annie, who has a level 41 Ivysaur and a level 42 Weezing. You get 4200 Pokédollars for winning.
Go southeast from the leftmost Tamer to reach the Gym Leader.
Gym Leader Koga
Gym Leader Koga has a level 43 Weezing, a level 43 Muk, a level 43 Golbat, and a level 44 Venomoth. You get 7040 Pokédollars for winning. After the battle, you get a Soul Badge. This increases the maximum level at which traded pokémon obey you. You also get TM27 Toxic.
After the battle, the invisible walls in the gym will be gone.
If you have a level 45 pokémon now and you couldn't get into the Saffron City Gym before, go there to get another gym badge when you are ready.
Things You Can Do Next
There are several new places that you can reach after you learn Sea Skim in Fuchsia City.
You can go east from Cerulean City and surf in the water in Route 10 to reach the Power Plant, an optional area with a rare pokémon at the end.