The tunnel that connects Vermilion City to Route 2 is called Diglett Cave. There are nothing but Diglett and Dugtrio throughout the path. Both are incredibly useful against the Vermilion City Gym Leader.
Route 2 – East Side
If you don’t remember, Route 2 connects Viridian City with Pewter City, and has the Viridian Forest along the west side. The east side can be reached through Diglett Cave.
To the south is a small house with two people inside. One will trade Pokemon with you. If you have an Abra, he will give you a Mr. Mime. This is the only way to get Mr. Mime in the game, but you can always return at a later time.
Exit out of the house and continue south. You will need to use the move Cut to open the path further south. Another building will be along the route. Professor Oak’s Aide will be waiting inside this building. If you’ve captured 10 different kinds of Pokemon—and you should have by now—he will give you HM05 Flash. Flash allows you to illuminate completely dark caves such as Rock Tunnel. This isn’t necessary, but it makes the game a heck of a lot easier.
Keep going south along route 2 to find an HP Up and a Moon Stone. That will clear out Route 2. Cut a tree to the west and follow the path south to Viridian City.
On the far southwest corner of Viridian City is a small tree that can be cut down. On the other side of the tree is a sleeping man. Wake this man to hear about his dream. He will then give you TM42 Dream Eater. This move is a psychic-type attack that will drain a sleeping Pokemon’s HP and give a portion back to the user.
Pewter City Museum
The final side quest is located in Pewter City. Now that you have HM01 Cut, enter the Pewter City Museum from the back door on the right side of the building. Speak with the scientist in the back to receive Old Amber. You won’t be able to do anything with this right now, so deposit it in the PC.
With all these side quests out of the way, return to Cerulean City and cut the tree to the east. This will open up Route 9.