Places to See with the Kids: Palaces
Toksugung Palace, smack dab down in central Seoul, is a great place for the kids to run around a bit in nice weather.
There are also small ponds to see with turtles and big gold fish. In the early evening there is a changing of the guard ceremony, with traditional Korean uniforms in abundance.
The grounds at Toksugung also host two art museums: one with classical Korean art (free once you've paid to enter the Palace grounds themselves) and the other with rotating exhibits (separate admission charge). There are exhibits of armor and swords in the first museum that will interest some older children, but otherwise there is not so much for younger kids to see. Maybe if you are two parents you can take turns, as the museums are not too large. Take the Red Line subway to CITY HALL station.
Seoul's other major urban palace, Kyongbokgung, offers a nice garden with a pagoda-like structure in the middle of a large pond. The view from the Palace of the mountains to the north is wonderful on clear days, with snow caps in the winter. There is a nearby Korean Folk museum (more interesting to older kids) to see, and you can often catch a glimpse of Korea's Blue House (the presidential mansion) to the north from the Palace grounds. Take the Yellow-Orange subway line to ANGUK station