There is much to do in Cerulean City. You will have to battle your Rival, beat a rain of trainers, stop Team Rocket from their usual deeds and of course earn another badge. The way out is blocked by a guard, who will move away after you visit Bill at his house on Route 25.
When you go north, attempting to go on the bridge to Bill's house, your Rival will stop you for a fight. After the battle has ended, he will give you a Fame Checker.
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You might want to wait with the gym battle after you've visited Bill, since Misty can be a tough opponent at this point of the game. After you beat Misty you will be able to use the HM01 Cut, outside of battle.
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At the house west to the PokéCenter there is an old man who will trade Pokémon with you. You will get a Jynx at the same level as the Poliwhirl you have given.
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You may buy a bike from the shop at the southwest corner of the city. However, it costs
1,000,000, which is too expensive for this part of the game. Even if you do have money to afford these, there isn't really any point in buying them because you will receive parts later on the game which you will exchange for a bike in this shop for free.
When you finally leaving Cerulean from the house where the guard used to be, a Team Rocket Grunt will stop and fight you. When you defeat him he will give you a TM28 Dig.
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