South Korea is a relatively small country and domestic travel is both cheap and convenient. With more young Koreans spurning the traditional package tour complete with transporation, hotels, and guide, hostels have become more and more common in Korea over the past decade.
Jeju hostels
Jeju is the premier vacation spot in Korea and there are several youth hostels on the island. However, most of the hostels do not have English websites or English speakers to answer phone calls. Here is one:
Here are some good links for information about traveling aroung South Korea:
Some websites for info about Korea tours in English:
There are three airlines in Korea:
COMMERCAIL AIRLINES
BUDGET AIRLINE
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KTX/Korail
Here is the website for you to find timetables/prices/book tickets/etc for KTX or Korail! Be sure to check prices because to some cities it's a huge difference in price, and others, it's hardly anything!
http://english.barota.com/
Although these are faster, the express buses are also great and VERY comfortable! Check with your city to learn more about your station. Be weary that these can have big backups with traffic though!