You require a Bike Voucher in order to obtain a Bike in the Kanto Region.
You can find one in the Vermillion Speak to the Bike shop owner in Cerulean to start the quest. Then head south to Vermillion city and talk to the chairman of the Pokémon Fanclub and he will give it to you the voucher.
If you go to the bike shop in Kanto, located in Cerulean city,
he will tell you that unless you have a bike voucher, you will not be able to get a bike.
Where can we get the bike voucher?
If you go to Vermilion City and you head to the south, there is a row of 2 houses, one of them is the Pokemon Fans Club.
If you talk to chairman Charly and answer his questions, he will give you a bike voucher,
now all you have to do is go back to Celulean Bike Shop and exchange it.
Going to the bike Store in goldenrod city, speak to the store owner to get given a bike, no voucher needed