Distances in Phuket are not too big. From the north tip of the island to its southern tip, it can take you 45 minutes by car; and from the east coast to the west coast it can take you half an hour.
Roads are in good condition, and traffic is moderate. The best way of moving around, and the safest, is the taxi.
Many foreigners rent a motorbike, but I would only recommend it to people used to riding motorbikes in unfriendly environments, and for short distances. Everyday there are accidents, some fatal, with foreigners riding rented motorbikes involved.
If you do not want to take taxis or bikes, in Phuket there is a service of public buses or collective taxis, which consist of modified pick-ups. Most of them follow a fixed route along the main beaches. They are not very punctual or reliable, but they are a cheap option.
♥ Visit Tagulandang Island ♥