Vermilion City is a small town in central Kanto. The Pokemon Center is located directly to the west of the entrance, and the Poke Mart is located to the south of the entrance.
After healing up and buying some items, you can visit the fishing guru in the house to the west of the Pokemon Center. Speak with the man inside to obtain the Old Rod. Use this on bodies of water around Kanto to find Magikarp. Magikarp is a pretty lame Pokemon, but it evolves into the powerful Gyarados at level 20.
Enter the building two houses to the west of the Poke Mart. This is the Pokemon Fan Club. Speak with the Fan Club’s president to hear about his favorite Pokemon. He will give you a Bike Voucher for your time. Take this Bike Voucher to the Bike Shop in Cerulean City to obtain a Bike! This will help you get around much quicker.
The Gym is located to the south, but you won’t be able to reach it until you obtain HM01 Cut. That item can be found in the S.S. Anne, which is at the docks. It’s best if you go there before going to Route 11 to the east.