A bus circles the island four times a day, stopping at various points. Locals just flag the bus down, so you should too. Due to its relative infrequency, the bus really isn’t an ideal way of getting around the island unless you’re absolutely committed to going with the flow completely. If you’ve got anything even vaguely resembling a schedule, you’re better off renting a scooter. Hitch-hiking is also an option.
There’s a vehicle rental shop next to the Lanyu Hotel. Motorcycles cost NT500 a day and cars NT1500 a day, excluding petrol. In winter, renting a car might be safer and more comfortable than a motorcycle because of the slippery road conditions.