The kid in the house near the Pokémon Center is looking for a Bellsprout. If you have one to spare, it's a good idea to trade it to him. Rocky, the Onix you get in return, can make the Gym battle here much easier.
Earl's Pokémon Academy
The Academy is where young prospective Trainers learn how to raise Pokémon. On the blackboard are descriptions of the several status problems that can affect a Pokémon.
The Sprout Tower is a towering pagoda run by Buddhist monks. Since these monks also train Pokémon, this is a good opportunity to train before challenging the Violet Gym. Rattata roam the tower during daylight hours, and Gastly can be found at night. After battling past the others, you find your rival having just defeated the head monk. He talks to you briefly before using an Escape Rope. After defeating the elder monk, Sage Li, he rewards you with HM05 (Flash).
VIOLET CITY POKéMON GYM LEADER: FALKNER
The Elegant Master of Flying Pokémon
The Violet Gym specializes in Flying-type Pokémon. Clip their wings with Electric attacks, or knock them out of the sky with Rock moves. Grass and Bug Pokémon, beware; Gust is now classified as a Flying-type attack, inflicting double the normal damage. The Onix you just received will be helpful, but watch out for Mud-Slap, a Ground-type move.
Professor Elm calls when you step outside, asking that you visit his assistant in the Pokémon Center. The assistant delivers the Mystery Egg to you, as it must be near an active team of Pokémon to grow and hatch. Call the professor when it finally hatches, and he asks to see it; after inspecting the newly-hatched Pokémon, he gives you an Everstone as thanks.
With the Violet Gym defeated, head south to Route 32.