There are several different airlines you can fly with from Gimpo Airport (Seoul's domestic airport) to Jeju, including the 2 big guys, Korean and Asiana, as well as Jeju Air, Air Busan and probably one or 2 other discount airlines.
I would then suggest flying from Jeju to Busan (Gimhae Airport) as it's the closest airport with a large choice of flights from Jeju (you could fly to Daegu instead, which is closer to Gyeongju, but there are fewer daily flights and they'd probably be more expensive). I usually fly Air Busan from Busan to Jeju and I've found that I can get tickets cheaper on expedia (the Canadian site, never checked the American site). From Gimhae Airport there are non-stop buses every hour or so to Gyeongju taking a little over an hour.
Finally, from Gyeongju your best bet is probably to take a bus to Gangneung which are fairly frequent and take 4-5 hours. Then take a bus from Gangneung to Sokcho, which I believe leave every 20 minutes or so and take about 45 minutes to an hour.
I've been to all of those places, so if you have any more questions, just let me know.
Jason