Quests

Bike Voucher
New to Vermilion City, you accidently wonder into the Pokemon Fan. You ask an old man where the Pokemon Gym was, but instead, end up listening to a story about his Pokemon. After listening to his story, his gives you an reward.
Reward: 5 Trainer Exp, Bike Voucher

Cut Master
After recieving the invitation to the SS Anna party from Bill, you hear about the Captain who has the HM Cut. When you go to the Captain's Office, he sitting over a garbage can regergatating(puking). You comfort him. He then feels better and gives you the HM Cut. This is required to get to the Vermilion Gym.
Reward: 10 Trainer Exp, H.M. Cut

Captain Jack's Chest   REQUIRES Water Type Pokemon above Level 10.
In a house by the port lives a man named Captain Jack. He needs you and your Water Type Pokemon to swim down to the bottom of the Vermillion lake, and find his gold. If you succeed, he will give you 500 gold out of the chest. If you don't succeed, you will do the following: You will go underwater with your Water Type Pokemon, searching for the treasure. Instead of going down to the chest, you discover a coral reef. Swim through the reef, providing great detail on the amazing color and enriched stone engraving on the walls. After you and your Pokemon play with the Goldeen for a little while, one grows fond of you and decides to come with you.
Reward: 25 Trainer Exp & 500 Yen - OR - 20 Trainer Exp & Level 5 Female Goldeen
Requirements: Water Type Pokemon above Level 10 in your Party

Beached Wailord
In Vermillion, there is a Wailord that has washed up on the beach. You then talk to a girl there and inspire her to get her Wailmer friends to help puch the Wailord back into the water. Roleplay the Wailmer saving Wailord and the girl will give you her necklace as thanks for the idea, which turns out to be a Mystic Water.
Reward: Mystic Water