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 200 baht since the one-way street system was inaugurated.
For longer distances the metered taxis are generally cheaper.
Motorbike taxi rides within Patong start around 40 baht, though for any unwalkable distance you might find it hard to get a ride for less than 100 baht.