South Korea

  Gyeongju (loads of historical sites)
  Jeonju (old city)
  Busan (port city with great seafood)
  Jeju (nature)

–

  Palaces (Gyeongbukgoong, Changdeokgoong, etc.)
  Seodaemun Prison History Museum-history during the Japanese Colonial Period
  Insadong-there is a lot of traditional Korean food and cultural artifacts
  Bukchon Hanok Village
  Gwangjang Market-a lot of shops and street foods
  DMZ tour
  Hiking at Bukhansan

While you're in Jeonju, add Namwon to your lists of possibilities. (It's kind of far but in the general area.) Hyangiram in Yeosu is very cool, if you're up for a ton of stairs. 3

— Suncheon