Hi Jamie and welcome to Korea and KOTR! If you are living in Seoul, then you should have access to lots of climbing areas and gyms. Just click the gyms link on the left scroll bar and see what is near you. There are a lot of KOTRers in Seoul, and if you contact one of them you might find yourself in the same area and going to the same gym. Try sending a pm to Les (Les), Gabe (gt29905), Jason (dehaan), Patty (pattyannblake), Heidi (dangerflower) or Dan (dan13). If you need to do any gear shopping the climbing shops around Jongro-5-ka (종로5가) are cheap and plentiful.
There will be heaps of climbing happening every weekend, all over Korea and beginner or expert I'm sure you'll find both the expat and local climbing communities really open and friendly. Try these KOTR links for some more help too:
Finally, there is a group of folks heading over to Ganhyeon crag this coming weekend (Sept 4-6). It's sport climbing with super convenience, good camping/ min-bakking and a great atmosphere. Bus and trains from Seoul area go there. Just check out the link:
Hope that helps you out. Oh, and also do fill in your profile page a bit, as it helps others help you with regards to partners, gear and grades etc.