Because Misty's Gym is empty, take the path North across Nugget Bridge towards where Bill's house was.
You'll encounter 6 trainers who will challenge you to a fight, and promise a prize for winning. Once you get done with 6, there's a 7th... He said he wanted to fight you because you'd be tired... All he has a Porygon anyway, so he's not tough.

Continue on toward where Bill was and you will find Misty romancing with someone. She'll run off back to her Gym after you see her.
Bill's House has his grandfather in who will give you stones if you show him particular Pokémon in This Order:

Lickitung - Everstone
Oddish - Leaf Stone
Staryu - Water Stone
Vulpix/Growlithe - Fire Stone
Pichu - Thunder Stone

He will only give you one so use it wisely. Now Head Off To Misty's Gym back in Cerulean:

Misty is still using Water Pokémon. They can be easily taken out with a few Electric and grass attacks

Misty	Golduck Lv. 42	Quagsire Lv. 42	Lapras Lv. 44	Starmie Lv. 47

After beating Misty she will give you her Cascade Badge and you should be on your way. Head to Lavender Town now for some sight seeing

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