After battling Team Rocket in Mt. Moon and getting a fossil, you go east to reach Cerulean City.

Explore the City

In the house east of the entrance, there is a woman who will give you a Bulbasaur if you have caught at least 30 pokémon.

East of that, there is a fountain that you can check on and throw money into. If you throw money in, your pokémon will become happier.

A man in the pokémon center can teach your starter pokémon a special move. You can choose from a water attack, an electric attack, or a fire attack. If your starter is Eevee, the electric attack could be helpful in the next pokémon gym.

There is a girl in the Pokémon Center named Tatianna who will give you a level 12 Alolan Rattata if you give her a Rattata.

Go into the northwest house, then go through the back door and go east to find a Rare Candy.

There is a bike shop in the southwest part of town. If you look at all the bikes in the shop and listen to the Bike Maniac talk about them, he gives you 5 Heart Scales.

If you try to use the north exit of town, your rival will challenge you to a battle. He has a level 12 Pidgey, a level 12 Oddish, and a level 13 Pikachu (if you are playing Let's Go Eevee) or a level 13 Eevee (if you are playing Let's Go Pikachu). You get 520 Pokédollars for winning.

After the battle, your rival talks to you, then runs away.

When you are ready, go to the right of the Pokémon Center and enter the Cerulean City Gym.