In the house next to the Pokémon Center, an old man is willing to trade his Ice/Psychic-type Jynx to get a Poliwhirl. Poliwhirl can only be found by using a high-end fishing rod, so you will need to return later to complete the trade.
Gym Badge Man
In the northwest house is a man who will explain the effects of the eight Gym Badges. The first five Badges enable the use of HM moves in the field. Four Badges increase a certain stat during battle, raising each respective skill to 112.5% of the normal value. The four other Badges ensure a Pokémon's obedience during battle.
The Bike Shop has bicycles for sale, but the price is a whopping 1,000,000! You can't carry that much cash, so it seems that you are out of luck for now.
The house behind the Gym is blocked off by a police officer. It seems that Team Rocket has stolen a valuable item from the people living here. Nothing can be done about this now, so set your sights on the Gym.
CERULEAN CITY POKéMON GYM LEADER: MISTY
The Tomboyish Mermaid!
Defeated, Misty awards you the Cascade Badge, which ensures your Pokémon's obedience up to level 30, even those you got in a trade. It also enables the use of Cut outside of battle. Misty then presents you with TM11 (BubbleBeam) as a prize.
As soon as you head for Nugget Bridge, Blue appears. He taunts you like usual before challenging you to another battle. His Pidgeotto is more a nuisance than a threat. Abra, with Teleport—which fails during Trainer battles—as its only move, is unable to deal any damage to your Pokémon, giving a good opportunity to level up your weaker Pokémon. Rattata's Hyper Fang can inflict serious damage, so take it out quickly. His starter Pokémon has learned a damaging elemental attack, so avoid pitting your own starter against it.
Upon defeating him, there are sixteen Trainers between Cerulean City and the Sea Cottage; luckily you can backtrack to the Pokémon Center to heal up as necessary. When you beat the five Trainers on the bridge, a disguised Rocket will give you a Nugget. He then tries to get you to join the organization, battling when you refuse to join.
A patch of tall grass is located near the north end of the bridge. In addition to the Jr. Trainer♂, wild Abra can also be found here. These Pokémon tend to teleport away as soon as battle begins, so try having a speedy Pokémon put it to sleep to prevent this. An extra Abra will be useful later on, in a trade for another Psychic Pokémon. Don't miss TM45 (Thunder Wave) on the hill to the north. Thunder Wave is a non-damaging move that causes Paralysis, a helpful tactic when attempting capture.
There are nine more Trainers to fight on your way to the cape. One of them is standing in a line of trees, and you'll need to lure him out with a battle to reach TM19 (Seismic Toss) lying behind him. Seismic Toss can be helpful against durable Pokémon because it inflicts damage equal to the user's level, regardless of the combatants' Attack or Defense.
The Sea Cottage on Kanto Route 25 belongs to Bill, a famous Pokémon researcher. His latest experiment has gone wrong, and his DNA was combined with that of a Pokémon! He asks for your help, so when he steps into the machine, activate the program on his PC to return him to normal. As thanks, he gives you an S.S. Ticket. This ticket lets you board the luxury cruiseliner that is docked in Vermilion City, which also happens to be the location of the third Gym. Before leaving the cape, re-enter the cottage and check out Bill's PC to view his favorite Pokémon, the Eevee family. This adds threenewpages to your Pokédex.
Back in Cerulean, the officer blocking the burgled house has moved out of the way. Inside lie obvious signs of a break-in, including muddy footprints, overturned plants, and a gaping hole in the wall that was used as an escape route. Step through the hole to find the thief.
The house on the hill is the Pokémon Day Care. The old man inside will raise one Pokémon at a time while you are away. A Pokémon left here will gain one experience point for every step taken. If you bought the Magikarp near Mt. Moon, this is a good way to raise it. The downside is that he will make every decision; he may delete a move you like, or ignore a new move. He charges 100, plus another 100 for every new level, to return a Pokémon.