Getting to Jeju
You can get a to Jeju flight from Kwanju airport one way for about 60,000 and it takes about 30 minutes. If you take the ferry from Mokpo, it'll cost you 30-44,000W one way and it'll take 4 and a half hours. I'm totally for flying personally. Also, there's a ferry from Busan, but I'm not sure about the price. I would imagine it's more expensive. And flying from Busan is probably about the same amount. If you're planning on going to Jeju very often, it's gonna be a substantial chunk out of your pay check, no matter how you slice it. Day trips are possible but only if you fly, and whether you go by ferry or plane, you MUST book your tickets in advance.
If you just want to travel around a lot, I'd just look for a place that has good mobility. Somewhere that has a KTX station and maybe an airport nearby. So you could try Incheon (about 30-40mins from Seoul), Kwanju(Jeollabuk-do), Busan. Busan (gyeonseong-the little bright green one in the south) is really beautiful, but I'm not sure what the climbing is like...if it's easily accesible without a car, and what the gyms are like. If you go to Busan, I recommend trying to land a job near Haeundae beach, or Jansang area. But I've heard Gyeonggi-do is the place for climbing. All these locations are probably confusing. Check out the map "climbs by province" and then you'll have a better idea of where things are.