I’ve heard the National Museum of Modern & Contemporary Art is a can’t-miss, as is Gyeongbokgung Palace, the 14th Century dwelling of the Joseon Dynasty. Try to book a stay of at least one night in Bukchon Hanok Village, a traditional area not far from the Palace. Nearby: Kyung-In Museum of Fine Art, where you can also have tea & snacks under outdoor pavilions. The National Museum of Korea is excellent (Yongsan neighborhood). The most important Buddhist temple in Seoul is Jogyesa; if you’re into temples, another is Bongeunsua in south Seoul. Finally, you MUST catch a baseball game: 3 KBO teams play in and around Seoul. Public transit is amazing in Seoul. Buy a T-money or Cashbee card from a convenience store, and top it up with cash.