Adam's bike takes him through almost his whole journey. In fact, it's the first image we encounter in the entire book, and it continues to come into play in a bunch of different ways throughout the story.
Adam is constantly talking about pedaling his bike and the effort he puts into it. He's so involved with this process that as he begins he journey, he remarks, "I am the bike and the bike is me" (3.37). The bike is also central because of Adam's constant fear of its getting stolen. Paranoid? Oh wait, it actually happens.
So what does this all mean? The bike is a dynamic symbol, meaning it can be interpreted in many different ways. On the one hand, we associate it with the freedom Adam feels on his journey. He chooses to take the bike instead of the train so he won't feel "confined" (1.6). When he's on his bike, he is taken by momentum and feels reckless and courageous. On the other hand, we associate the bike with Adam himself. The bike can't always be controlled, like when it's traveling down a steep hill. This is also the case for Adam, who doesn't have control over his own life. Before it was controlled by Mr. Grey, and now it's controlled by doctors, medicine, and Brint.
Adam's feeling that the bike represents freedom is amped up when it gets stolen. He's really anxious for a while there, and once he gets the bike back, he's "sailing, sailing" and "going, going" (27.14). The doubling of the words here helps emphasize the movement and freedom Adam feels on his bike.
Finally, we have to remember that when Adam returns to the hospital, he looks outside and says, "Someday I will ride my bike out there" (31.8). Being away from the hospital would mean freedom. When he pictures that, he pictures his bike.