If you want to teach trad and anchor building, there are some easy routes near Seoul on Insubong
우정 B 인수 A Chouinard B (put up by Patagonia founder Yvon Chouinard when he did his military service here in the '60s)
Insu B are all multi-pitches with slab and cracks (of varying quality and style) that are 5.6-5.8. There are a few more, I think, and many more as you go up in grades (these are old school grades though!!). On nice weekends, Insu can get way crowded, though. Seoninbong, likewise, has niumerous climbs, but can also get crowded. The ridges at Daedunsan offer trad possibility. Seoraksan has great multipitch alpine-style routes. . . A2O and the ridge of Janggeunbong (aka "The General") and Yuseondae are beautiful and can teach something about routefinding too! (You should register in advance if you plan to climb at Seoraksan-- it's free and easy-- have the passport number for every foreigner in your party).
I'm sure there are less obvious places that include everything you want, and if they come to mind I'll let you know.
If you just want sport routes, Kanhyeon is awesome, everything Danielle says and more. There are even "practice" anchors (well, 2 bolts) on the ground so you can talk about equalizing etc.
Jay, there are lots of opportunities to learn/review techniques-- watch "upcoming trips"! You should head up to Seoraksan for Chuseok.