I'll move this info here:
Direct busses to Seonunsan run from Gwangju, Jeongeup and Gochang. Busses leave Gwangju for Seonunsa (the temple is Seoununsa. . . so the busses may be labled as such!) at 8:50, 10:20, 11:50, 1:15, 2:50 and 4:15.
From Jeong-eup it's 7:10, 11:00, 2:50 and 4:30. Busses head more frequently from both of these (and more) to Gochang, where busses leave to the park regularly.
Busses from Seoul Express Bus Terminal run to Gochang every 20-50 minutes from 7AM-7PM. Dongducheon people will probably go through Seoul. The train search wasn't working just this minute, but I think the trains from Seoul to jeongeup leave from Yongsan station. (There are no trains to Gochang, but if you arrive in Jeongeup and need a bus to Gochang, the station and terminal are not too far.
From Changwon there are just two busses a day (2:10 and 5:00) to Gwangju, but you can piece together busses if these aren't convenient (and I checked these online, so the info may not be 100%).
There are 31 trains to Jeong-eup from Seoul everyday-- they leave from Yongsan Station, though, so if you're coming from Seoul by train, seach from Yongsan (the train-search link is found on the left on KOTR). . . if you get the KTX, it's a little over 2 hours.
There are also busses from Nambu Terminal to Gochang, but you'd best call 02)1330 (Korea's free help-line in English) for exact times.
"If you can't do something well, you might as well learn to enjoy doing it poorly."
-- from a de-motivational poster, but I find it oddly liberating