Update (7 Mar ’16): You can get the most updated Jeju bus route map here, at the Jeju Tourism Organisation blog. The JTO blog also has a pictorial guide on how to take public buses around Jeju.
I had read that travelling by public transport on Jeju island was not easy and travel forums were full of enquiries about full-day taxi services. However, taking one of those services sounded too excessive for a single traveller, as it would generally cost between 120,000 to 150,000 won (abt 150-180 SGD).
The other option was to take bus tours and YEHA Guesthouse offers 3 different routes, costing only about 79,000 won each (before a 10,000 won discount for guesthouse customers). However, further examination of the bus tour routes revealed that there were certain places I would not be interested in. In addition, the lunch-stops that were included in the tour packages were at non-halal places.
There is also a Jeju city tour bus with a hop-on hop-off format for a mere 5,000 won; however, that will only take you to attractions in Jeju-city, covering only a small part of the island.
My love affair with Seoul and Korea 