Hey there Dave and Family,
If you are in Seoul round late June be prepared it will be rainy season. It is actually quite monsoonal and rains most days in the afternoon. That said you might be OK, as Im nit sure when the worst part starts and ends. In Seoul the place to climb is probably Insubong...a horrendously crowded place on weekends but 25-26th should be OK as it is a Monday-Tuesday. There are a lot of routes there, mostly all slab, crack nat-pro trad climbing. Check out the link under Rock Climbing Areas, Insubong. It's easy to get to, though the walk in from wehere the bus stops will take about 40-60 minutes before you reach the start of the climbs, Hundreds of climb shops and so on at the park (Bukhansan National Park) enterance. These can often supply you with a route map, and advice. Sport climbing areas in Seoul are unknown to me ....but do a quick search again under rock climbing areas Seoul. Maybe someone else will post a reply here about that.
If you wanted a good place, not in Seoul but accessible by a train ride try Ganhyeon/ Kanhyun. It is sport. No cams, nuts or wires needed. 60m rope, a dozen quickdraws and you are good to go.
OK Hope you have a good time when you get here. Keep in contact and we should be able to hook you guys up, and meet.
quem deus perdere vult, primus dementat