Tuk Tuks, taxis, songtheaws, scooters, scooter taxis or renting a car, transportation is available in any form and everywhere where there is a tourist in Phuket. Arriving at the airport countless taxis will be waiting and competing to transfer you to your resort in Phuket. Insist on the driver turning on the meter, especially for short distances to locations such as Nai Yang or Mai Khao instead of paying inflated prices. If you can negotiate a fixed fee for two persons included to locations further away such as Patong, Karon or Kata for 800 Baht or less you have a good deal though. Transportation on the island are plentiful and cheap.....if you know how to negotiate. For short distance travel, Tuk Tuk's are a fun way of transport, but you should be able to get to any destination within a 5 km radius for 150 Baht or less. Songtheaws, which are basically a pick-up truck converted to have two benches in the back is maybe even better, cheaper, the safest and also more "local".
If you want it really cheap, rent a taxi scooter. They are easy to spot as all drivers wear a colour coded and numbered vest. Fifty Baht should get you where you need to be. Renting your own scooter will give you much freedom to come and go as you please from around 200 Baht per day, but be ware. Getting your own scooter does not come without risk and, unless you really know how to drive and willing to negotiate others who often obviously don't know, rather choose a different option. Firstly, if you do rent a scooter, make sure you have an international driver license, get proof of the scooter's insurance and take pictures of any scratches or damages prior to taking possession. This will eliminate the possibility of problems once you return the vehicle. Many hotels in Phuket also rent out their own scooters and renting from them directly might be a good idea. Secondly, always wear a helmet and drive slowly, especially in wet weather. Phuket sees too many tourists getting seriously injured and even die due to scooter accidents on winding roads and steep hills. There seems to be a culture of being cool if you drive without a helmet. Do NOT follow this example.