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. The rates are generally much higher than in other places, but the vehicles are almost new and in excellent condition. 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 per day. The main road is sealed and 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. 4x4 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. Generaly most people just stay on the beach of their choice and walk to wherever they want to go.
Hitching on Ko Chang is also an alternative if you choose not to pay the often exorbitant fees of the songthaew. Many islanders are more than willing to pick up a hitchhiker who happens to be going the same way they are. A Coke or cold bottle of green tea for the driver are always appreciated at the end of your journey.