Jeju

I don't think it's even possible to travel around Jeju with a public transportation.

There arn't many public transportation there, and getting informations about it isn't easy.
(Even if you find a source of information, they buses only come like once a hour. So it won't be easy scheduling your trip)

Renting a scooter is cheap. Renting a car is cheap as well if you have a company.

Well have a nice trip!

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Oh. There is a tracking course in Jeju, called 'Jeju olleh'

It's newly made, but is like the hottest place to travel among Koreans now.
It's very long ( 22 courses), so you can't walk the whole course, but walking some of it might be cool.

I don't think it's mentioned in lonely planet book yet since it's new.

But you should really try it!

Here is the offical webpage.
http://www.jejuolle.org:8080/eng/

You can definitely take a bus from Jeju City to the south end of the island, but it may be best to rent scooters for travel between the lava tubes, the beaches, and Mt. Hallasan. I would recommend staying at Jeju Hiking Inn.

Post #1 is wrong, publis buses around Jeju are fine, they may not be to frequent but the najor routes around the island and too major attractions can be taken easily.