I've lived on Jeju for 3 years and will be leaving for good next week. Ralf at Big Blue 33 has the only fluent English speaking divers on the staff on the island - Ralf is German and lectured in languages at Jeju University before he set up his shop. He has several ex-pat divers as dive masters too who live on the island.
Jeju is a 1 hour plane ride from Seoul- I think EastarJet has the cheapest flights, but check the Korea Flight app for domestic flights in Korea.
The diving in Jeju is incredible. The waters off the south side of the island tends to be a degree or two warmer than the north. At Little Munsom, an island 5 minutes off the coast of Jeju, you will have a totally different diving experience depending on which direction you take from the entry point. Soft corals, kelp forests, a whole range of fish, obviously depending on season. On my last dive before I packed up my gear I saw cardinal fish laying eggs, a huge grouper (as well as a couple of smaller ones) frog fish, cuttle fish, scorpion fish some flounder and a ray!
Between September to early November I use a 5mm wetsuit. From December to now, I put a 10mm shortie over the top, but others use a drysuit. In fairness, this month I could have probably done without the shortie- the water was 20 degrees on my last dive. The fishermen's councils which tend to be controlled by the Haenyo- traditional women skin divers - are fiercely protective of their territory and some waters are off limits to recreational divers.
Ralf's prices are good and he has a fully equipped shop so you wouldn't have to take all your gear on the plane if there were tight weight limits. The ex-pat divers who use his shop are all good fun and several live locally, so visits to Jeju might be worthwhile for you.
Sorry we won't get to meet, but enjoy it in my absence!
---------- Post added June 19th, 2014 at 11:03 PM ----------
Sorry, forgot to say that diving on Jeju does tend to close down from late December until March.