Bike and Bike Voucher
In the Johto region, you will receive your Bike in Goldenrod City while progressing through they story line.
In order to get your bike in the Kanto region you must first get the Bike Voucher.
You will have to talk to the bike shop owner in Cerulean City to start the quest. Then you will have to travel to Vermillion city to get the Bike Voucher.
Once you make your way to Vermillion City, you will have to talk to Chairman Charly at the Pokemon Fan Club. The Pokeman Fan Club is a little south on the west side of the city. After you listen to Chairman Charly talk he will give you the Bike Voucher.
Once you have the Bike Voucher, head back to the bike shop in Cerulean City and give him the voucher and he will give you a bike.
NOTE THAT YOU CAN RECEIVE A BIKE “SKIN” FROM LOOT BOXES. YOU STILL NEED THE ACTUAL BIKE BEFORE YOU CAN USE IT. PRESS SHIFT TO USE YOUR MOUNTS.