In states with uniform bicycle laws, only the state can regulate cycling. Cyclists can (more or less) expect the same rules when traveling through the state.
Local jurisdictions can impose their own restrictions in states without a uniformity requirement.
California’s uniformity law (CVC 21) means cities may not ban bikes from local roads, although that hasn’t kept some jurisdictions from trying. San Jose International Airport, for example, persists in posting a “bikes must exit” sign on Airport Boulevard although they have no legal authority to do so.