Getting around Phuket is easy although may be expensive if you are in a weak bargaining position.  There are a few options available, and choosing one depends on a personal preference. 

Costing between 150 – 350 Baht/day, motor bikes are available for hire all over the island.  This is a good option when planning extensive exploring.  If choosing this option, make sure to inspect the bike for pre-existing damage before renting it, to avoid being charged later.  Another thing to consider before hopping on a motor bike is that driving is done on the left hand side of the road throughout Thailand.  Also, note that motorbikes are the greatest cause of serious injury on the island. 

Riding on the back of a motorbike taxi is another option for getting around.  This is the most inexpensive of the transportation options, but is not always comfortable.  Make sure to request a helmet before ever getting on a motorbike.  Not only is this for safety, but police will charge 500 Baht for anybody breaking the helmet law. 

Other options for getting around are Tuk Tuks (three wheeled, open aired taxis), taxis, busses, and car rentals.  All of these are readily available and fairly inexpensive, with car rentals costing the most. You might want to use caution when taking Tuk Tuks around Phuket. They can overcharge and together in concert which means they have agreed a price and do not waver from it. There has been some incidents of some drivers being overaggressive to passengers.

Long tailed boats are the most common form of local water transportation. 

This is may be the best Phuket Map site right now, with some excellent static maps and video journeys.