For such a small island it sure has a lot to offer.
Obviously the #1 and #2 things are sight seeing and enjoying the beaches, but there's also a lot of cool things like:
Hot air balloon rides
Submarine rides
Scuba diving/Snorkeling/Surfing/Para sailing/Water-skiing
Rent a scooter to sight see
Hiking the volcano (Hallasan)
Spelunking in the Lava tubes
Shopping in Jeju City
Visit the Botanical Gardens
Soingook Theme Park
Seogwipo Pleasure Boat
Horse riding
Golf
Pheasant hunting
There's also a Large stadium that im guessing has professional soccer games.
Heres a nice little list also:
http://www.frommers.com/destinations/jeju-do/4033_indattr.html