Here you can get the Bike Voucher by visiting the Pokémon Fan Club. It will let you get the bike in Cerulean City for free. (like you are ever going to get $1000000...) There is also an old man who will give you an Old Rod, which could fish for Magikarps only. A third thing is that a boy will trade a Far'Fetched for a Spearow. After that, go to S.S. Anne to the south. You can get on the boat with the S.S. Ticket. On the boat, there are many trainers and items. You'll find your first Great Ball here. Gary will be here (again) with a different appearance. After you beat him you can talk to the captain. He will give you HM 01 (cut) after you help him through his seasickness. Then boat will take off when you exit. The HM is very important. It will now let you do things you couldn't do before, like going back to Pallet Town, or getting Old Amber in the back
door to Pewter City's museum. The Diglett's cave to the east has some high-level Digletts and Dugtrios. If you get them they will help you a lot for the boss. Now, teach HM 01 to a Plant-type Pokémon. The move is even good in battle sometimes!
Go to the southwest of the city, CUT a bush, and go to the Gym. The leader there has a electric lock door. You have to find one switch, and another switch which is always BESIDE it. Defeat Lt. Surge, the gym leader, who has a Voltorb, a Pikachu(!!) and a Raichu(!!!). Digletts and Dugtrios can defeat them easily, as well as anyone you gave your DIG technique to. Talk to him afterwards to get TM 24, which contains THUNDERBOLT, a powerful version of THUNDERSHOCK. One of the Prof. Oaks' aids is also in a building to the east. He will give you the Itemfinder if you have more than 30 Pokémon. This item finds items... duh. But not too important really. By him is a guy who will trade a Nidorino for your Nidorina. After this, go into the Diglett's cave again. It will lead you near Pewter city and you can do things you couldn't do before. A person will trade a Mr. Mime for an Abra. So if you have a spare one, trade it. (This is the only way to get a Mr. Mime) You can also get HM 05 (flash) from one
of Prof. Oak's aides if you caught more than 10 Pokémon. Then go back and
up to Cerulean City. From there go east into the Rock Tunnel.