Probably the cheapest and most convenient way to get around is to rent a motorbike for yourself which is suitable for one or two people. These can be rented at various shops around the island, with one located a few hundred meters to the east of the traffic circle on Patak Road East. Typical rates run 150non-automatic to 200 Baht for automatic shift models. This would allow you to get around anywhere on the island for one flat rate for 24 hours. Be careful with this though as driving up and down the winding mountain roads is not necessarily for the first time driver.
Another option to get around is by tuk-tuk or taxi, which are mostly needed to go to other parts of the beach or nearby towns. To get to Phuket Town taxis are available and cost around 400 baht. There are also public busesSongthaews which leave from Karon to different places every hour for around 20 baht.
Karon - Patong will cost you 300 baht per tuk-tuk ride - though if you keep moving through the drivers you will eventually find one for 250. The local tuk tuk mafia seem to have stitched up the prices. From Patong - Karon Beach should be cheaper as there are a lot more tuk-tuk trying to pick up passengers.