Motorbikes can be rented everywhere. Rates are approx 200 Baht for 24 hours for a small moped, and up to 1000 Baht / 24 hours for a big bike. Moped are the most common form of transport on phuket.
There are thousands of accidents every year on phuket which mostly involve motorbikes and the death toll is very high:
(1) Wear a helmet - it's the law, or face a fine. (2) You need a driving licence; the rental companies won't ask for it, but the police will if you're stopped. (3) There is NO insurance coverage. If you or the bike are damaged you will have to pay for everything. (4) Make sure the owner is aware of all the existing damage to the bike before you start riding it.