Take a Tuk Tuk

...a typical Phuket Tuk Tuk

These are the other "public transport" around town and the island, BUT be warned they are not cheap especially in the main holiday areas. For long journeys it's often cheaper, and more comfortable, to take an air-conditioned taxi.
The Tuk-Tuk is a small open-air truck which, depending how the seating is arrranged, can carry between 6 - 10 people. They may look "quaint" from the outside but due to the roof design most people can't see out without permanently bending down. When bouncing over pot-holes in the road it doesn't make for a pleasant journey.
Tuk-Tuk's do not use meters but are "supposed" to have a chart of fixed fares for set routes/areas in which they operate, with other routes being negotiated between passenger and driver. In reality hardly any Tuk-Tuk's have a fare chart in them, or if they do then it's hidden away in some obscure place in the small drivers compartment where you can't see it. Therefore you MUST NEGOTIATE THE FARE for the journey you wish to take BEFORE GETTING ON BOARD. This is not only good advice but imperative as Tuk-Tuk drives a have a well deserved reputation for trying to charge highly extortionate fares, especially for short trips.
To take anyone within Patong Tuk-Tuk drivers will usually demand 100 Baht (however short the journey) even though the official fare is only 40 Baht. If you argue then many drivers will be abusive to you , or just drive off, so be prepared for it.
When it rains they immediately "DOUBLE" the normal fare they're demanding so travel around Patong by Tuk-Tuk suddenly becomes 200 Baht a trip.
Luckily the Tuk-Tuk drivers are not representative of the majority of smiling, courteous, happy and helpful Thais You meet all over this beautiful country.