Vermillion city

The house at the north-western corner has a Fisherman who'll give you an Old Rod, which allows you to catch Magikarp. Go south a bit, and you'll find two houses to the left. One of the houses is where you can trade a Spearow for a Farfetch'd. The other house is the Pokemon Fan Club. Talk to the chairman inside to get a Bike Voucher.

In the Pokemon Center, talk to the girl standing in front of the counter for a VS Seeker. This device allows you to rechallenge trainers who have trained their Pokemon to be even stronger than the first time you battled them. To activate the VS Seeker, simply stand around a couple of trainers and select the item. Trainers who are hoping around are the ones who will battle you again.

Now go back to Cerulean City, get yourself a bike from the Bike Shop, and come back in style!

All Aboard The S.S. Anne!

The S.S. Anne may seem small, but it's huge inside. It can be found by going south of Vermillion City, and "flashing" your S.S. Ticket to the guard. If you explore each and every room inside, you'll find tons of trainers and quite a few items. A girl in the second room from the right on the main deck can heal your injured Pokemon. In the ship's kitchen, you'll be able to find your first Berries, three of them to be exact, in the trash cans. From now on, all Berries will be stored in your new Berry Pouch. To end your stay at the S.S. Anne, fight Gary on the second floor.

After beating him, go up to the captain's cabin. He's seasick and puking. Rub his back and he'll give you HM01 - Cut. Now leave the ship. When you're outside, the ship will sail away. Remember: if you don't get Cut, the ship won't leave. If you want the S.S. Anne to stay, lose to a trainer after getting Cut. Now teach Cut to one of your Pokemon, and you're ready to go to the gym.

A Shocking Battle

Cut the tree blocking the path to the gym and go inside, where you'll face a few trainers who use Electric Pokemon. Now comes the hard part: Lt. Surge, the gym leader, has put electric locks on the door that leads to him, so you'll have to solve a puzzle. There are a bunch of trash cans. Press "A" on every one of them until it says that you've found and flipped a switch. You need to flip two switches for the door to open, so you'll need to find the second switch. The second switch is always either is always either above, beside, or below the first one. If you get it wrong, the switch will be reset, and you'll have to do everything over again. This is a very tricky and sometimes annoying puzzle, but hopefully, you'll get it sooner or later. And when you do open the doors, it's time to fight Lt. Surge.

Lt. Surge has the following Pokemon: a lv. 21 Voltorb, a lv. 18 Pikachu, and a lv. 24 Raichu. Beating him with a Ground-type Pokemon is as easy as tossing a PokeBall, and catching one is easy too. You can catch a Diglett or even a Dugtrio in the nearby Diglett's Cave. If you beat him, he'll give you TM34 - Shock Wave and a Thunder Badge.

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Diglett's Cave

If you go east from Vermilion City, you'll find Diglett's Cave on Route 11. This cave is no maze. It's simple to go through since there's only one road. There are no items to pick up. The only Pokemon you'll find are Diglett and the occasional Dugtrio. Head inside and go to the other end to go back to Route 2 for some important business.

Locations (Diglett's Cave)
Diglett Dugtrio

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Back On Route 2

After you exit the cave, you'll revisit Route 2. But this time, it's the other half of the route that you weren't able to explore before. Go south, and you'll find a house. Inside, a guy is willing to trade his Mr. Mime for your Abra.

The next house has one of Prof. Oak's aides in it. If you've collected ten or more Pokemon he will give you HM02 - Flash. It's used to illuminate dark caves and tunnels. Keep on going for two items.