Pokémon Gold and Silver/Cerulean City

'Head directly to the Gym to find the suspicious man who has run off with the Power Plant's part. The Gym Leader isn't in the Gym, but that's OK—you have other business to attend to before you battle with her.'

Apprehend the thief
The thief is hiding from the law in the Gym, and he'll run for it when you come to visit. Fortunately, the man outside saw which way he ran: north towards. Catch him on the bridge, defeat his lone Pokémon, and he'll fess up. He stashed the Machine Part in the Gym, and you'll find it by searching above the square of water in the center.

Return the stolen part
When you return the part to the Power Plant, you'll be rewarded with TM 07 (Zap Cannon). And when that's all over, you'll be able to recover the Copycat Girl's Clefairy Doll, which you can return to get the Pass which lets you ride the train from to  at any point, and pick up the Expansion Card in. You can use that card to wake up (see  for more details).

Something in the water
A kid near the upper left corner of the city is reporting a ping from his Item Finder, but he didn't follow it up. Head into the water and use your own Item Finder: You'll find the Berserk Gene which, when held, raises the Pokémon's Attack, but also Confuses it.

Find Misty and her mystery date
Once you've returned the Machine Part, you can come back to Cerulean and hunt down Misty. She's shirking Gym duty... to go on a date? You'll need to find your way through, but you'll find her and her mystery man at the end. When the guy dashes off, Misty will return to her post at the Gym.

Cerulean City Gym
Misty prefers Water-type Pokémon, but some of her creatures have dual-types like Water and Ice, or Water and Psychic. You have plenty of ways to deal with Water Pokémon by now. You can electrucute them, Grass them, or better yet, both. You'll definitely want a Grass-type (or a Flying-type) to deal with Misty's pesky, Electric-resistant. The can heal itself, so save a strong Pokémon to finish it off quickly.

Trainers

 * Swimmer♂ Parker
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 * Swimmer♀ Briana
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 * Swimmer♀ Diana
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Leader

 * Leader Misty
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* Corsola can only be found in the morning and daytime, while Staryu can only be found at night.