When you first arrive in Cerulean City, you will find a bike shop. However, the cost for purchasing a bike is extremely outrageous, making it impossible for a player to have that amount of money this early in the game. Because of this, you will eventually leave Cerulean City without a Bicycle. For now, you will be stuck walking from place to place.
A little later on in the Main Story, you will arrive in Vermilion City. Here, you will find the Pokemon Fan Club in one of the houses. By speaking with the Pokemon Fan Club President, you will receive the Bike Voucher. With this voucher, you can obtain a free bike from the bike shop back in Cerulean City!