After you take the train in from Incheong (around an hour), visit the Gyeongbokgung Palace, a truly awesome display of traditional Korean architecture complete with ornate buildings, rolling gardens and a large pond with a pavilion in the center. Catch the changing of the guard ceremony, which happens every hour between 10 a.m and 4 p.m.
Experience Seoul's vibrant feel on foot. Start at the Cheonggyecheon Stream, a popular promenade and recreational area surrounded by the city's modern skyscrapers on either side — you can relax in the shade or dip your toes in the cool stream on hot days. Or take on Namsan Mountain for some more vigorous hiking: Panoramic views of the city await you at the top.