Patong can just about be covered on foot, but there are plenty of taxis and motorbikes to get you around. The cost of these is one of the main down sides to Patong as they are unregulated and charge what they please.
Phuket has two types of taxi - millions or so it seems of small songthaew-style minivans usually bright red, occasionally bright yellow, and very few conventional sedan-style taxis yellow and red, with a "TAXI-METER" sign on top.
The minivans have no meter, and their drivers are notoriously mercenary, so always agree a price beforehand and do bargain hard. Short hops typically cost B200 since the one way system was invoked.
For longer distances the metered taxis are generally more suitable.
Motorbike taxi rides within Patong start around 40 baht, though for any non-walkable distance you might find it hard to get a ride for less than 100.