After you explore Cerulean City, you can enter the Cerulean City Gym and challenge gym leader Misty.
Show a Pokémon
The guy at the door will ask you to show him a pokémon that is at least level 15. If you don't have one, you should go back and battle wild pokémon until you have a pokémon with a high enough level.
Battle Trainers
After you get into the gym, go north and you will battle Beauty Lily, who has a level 16 Goldeen. You get 320 Pokédollars for winning.
Go north, and you can go to the left if you want to avoid a battle. If you don't go around the long way, you will battle Beauty Violet, who has a level 16 Shellder. You get 320 Pokédollars for winning.
Go north, and you can go to the right to avoid a battle. If you don't go around the long way, you will battle Beauty Daisy, who has a level 16 Seel. You get 320 Pokédollars for winning.
Battle Misty
Go north from there and talk to Misty. She has a level 18 Psyduck and a level 19 Starmie. You get 3040 Pokédollars for winning. After the battle, Misty gives you a Cascade Badge. With this badge, pokémon up to level 30 that you got from other trainers will listen to you. Misty also gives you TM29 Scald.
Now that you have the Cascade Badge, you can get Super Potions and Lures from the Poké Mart.
When you are ready to continue, go north from Cerulean City to Route 24.