Backpacking to South Korea

Hihi,

My friend and I will be planning a 2 weeks backpacking to South Korea in between April06 to May06, departing from S’pore to South Korea. We are planning to venture Jeju Island, Busan and Seoul.

The following are our initial plan
(1) By Air - S’pore to Jeju (which I think don’t have a direct carrier)
(2) By Air - Jeju to Busan (intent to travel via a local carrier)
(3) Busan to Seoul (via Korail and will be stopping over at some states)

Need assistance and advise to the mentioned places, on limited budget. My questions are as follows
(1) How many days should be spending on those mentioned places?
(2) Is there a budget carrier in Korea for Incheon to Jeju and Jeju to Busan? I browse through the website but the information are in Korean. Can we book or ticket online?
(3) Shall we engage a local guide for Jeju when we are there? I seen the promo packages for 3D2N and it costs KRW 670,000 / person, which I think is a bit costly and out of my budget. Where can I locate these guides with reasonable price (free & easy), all we need is the transportation to those highlight in Jeju.
(4) We are checking out the youth hostels in Busan and Jeju but managed to get only Guesthouse Busan and Jeju Hiking Inn from the website. Any other suggestion for the hostel or budget accommodations?

Please advice and thanks in advance.
Others suggestions and advice are welcome.

Sonel