In the daytime, you can catch a songthaew on its route around the main road for 50-200 baht/person (Feb 2020), 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 17:00, 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. 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 m to Dan Kao Pier.
Cars are also available for rent, most hotels can help with it. 4 x 4 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.
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.
Motorbikes are a fantastic way to explore the island. Small motorcycles can be hired for 150-250 baht per day. The main road is sealed and almost circles the island and there are plans to complete the circuit in the near future. The eastern road leading all the way to the War Memorial, at the island's southeasternmost point, is worth a trip.
Riding a motorbike is not for the inexperienced or faint of heart. Most visitors manage on level roads, during the daytime and during good weather conditions. Darkness and rain, together with poor skills, steep hills and questionable maintenance can be fatal. Think twice if you have to travel at night, and when it rains during the night, avoid the steep hills in the north and the southwest.
When renting a bike, check tires, brakes and light. Reputable shops will fix them while you wait, don't compromise on safety. Bring your own helmet if you have one.