For those who don't want to rent a car or motorbike, the bus is a perfectly viable option for a trip to Jeju. If you want to hit every sight it might be a bit difficult, but for those just looking for a relaxed beach vacation, getting on the bus once or twice a day should be a breeze. The intercity buses all accept your T-Money card, or you can pay in cash, usually between 1-2,000 won depending on how far you go. Most buses either start or end in Jeju City or Seogwipo and come every 20-40 minutes or so depending on the time of day and the route.
We took the bus from Jeju City to Hallim Park on Friday morning because of some problems getting a rental car. It took over an hour to get there, but only half an hour to get back. Not sure why that was, since there was no traffic either way. Sometimes you need to tell the driver where you're going so he can activate the T-Money card, so be sure you know the Korean name of your destination. I would highly recommend having some Korean under your belt before trying to navigate Jeju on your own as many signs, such as this bus terminal sign are only written in Korean.
Other options for seeing Jeju are renting a motorbike, car, bicycle or walking. Our car was the cheapest you could get, and it was 43,000 won per day. With a big group this could be a very affordable option, but you should double check the requirements for licenses. Lonely Planet says you need an international driver's license, but call the specific rental company for details. I never worry about these things since I just let the boyfriend drive.
We saw some people doing some serious biking around the island. If you enjoy biking and had enough time, I think a bicycle alone could be a viable option for exploring the island. In a car it takes about an hour to go north-south and two-three hours to go east to west. Muliply that how you like to figure how long it would take to bike.
Although you probably won't walk the entire island, there are many walking trails known as "Oleh-kil" (Jeju dialect for small street) which are generally clearly laid out just for walkers. You'll probably see more Oleh-kil in my future posts.