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 metres to the east of the traffic circle on Patak Rd (east). Typical rates run 150 baht (manual) to 200 baht for an automatic. 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, as well as the unpredictable Phuket traffic, is not necessarily for the first-time driver. A wise strategy is to take public transport for a short period to orient yourself, then hire a motorbike.
You can easily get around 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 buses (songthaews) which leave from Karon to different places every hour for around 20 baht.
Karon-Patong will cost you 400 baht per tuk-tuk ride, though if you keep moving through the drivers you may eventually find one for 350 baht. (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-tuks vying for passengers.