In the daytime, you can catch a songthaew on its route around the main road for 50 or 100 baht/person, depending how far you go. Starting from 5.00pm, many of them start to ask “taxi” price, telling you that they operate as a public transport only until that time, and may quote prices as high as say 500 baht from Lonely beach to the Dan Kao pier (bargain if you choose to hire, do not support silly prices!). However, if you have some time and patience, you still can try and have a “shared” ride with some drivers, maybe for a higher rate if they expect little or no other passengers.
These taxis are also waiting at the Dan Kao Pier (50 baht/person to White Sand beach, 100 baht to Lonely beach). At the Dan Kao Ferry-Pier there may be no taxis available. If you arrive without a vehicle you may have to walk the 400 metres to Dan Kao Pier.
Small motorcycles can be hired for 150-250B. The main road almost circles the island and there are plans to complete the circuit in the near future. Cars are also available for rental, most hotels can help with it. 4×4 car recommended, since some roads might be in bad condition, especially near Lonely Beach.
If there are 2 or more people going with you, hiring a songthaew may cost the same price, or even be cheaper than paying per each person in a “shared” songthaew (there is no difference, an empty songthaew can easily be hired). Just do not forget to bargain if their price sounds quite silly when compared, say, to Bangkok taxi-meter (on Ko Chang it can be difficult if not impossible to get the same price, but at least it should not cost double or even more). Most folk however, just stay put on the beach of their choice and walk to wherever they want to go.