Climb Bijobong peak, its the highest accessible point on the island and on a clear day can give you a great view of the island and the surrounding islands. It's a very easy climb under 2km on some paths and at the peak there is an observation deck made to resemble a wooden pagoda.
Camp at Batjireum Beach. It is the only way to stay on this beach, which is the islands nicest beach. You will have to bring your own equipment, but it is one of the nicest camping locations in the area.
Take a bus tour of the island. For 20000-30000 won you can charter a van for up to eight people to take you to all the islands major attractions, ask the owner of your pension/minbak if they can arrange this for you.