A first trip to Seoul, assuming you arrive the evening before Day 1.
Day 1 — Historic core
Morning at Gyeongbokgung (arrive by 09:30; closed Tuesdays), then walk to Bukchon Hanok Village and down through Insadong for lunch. Afternoon at Changdeokgung and its Secret Garden. Evening in Gwangjang Market for bindaetteok and mayak gimbap.
Day 2 — Modern Seoul
Start at Dongdaemun Design Plaza, then the shops and coffee of Seongsu-dong. Afternoon crossing the river to Gangnam or the Han River park at Ttukseom. Evening in Hongdae for buskers and late food.
Day 3 — Views and pick-your-own
Morning hike up Inwangsan or Naksan for the city wall and skyline. Afternoon free for a museum (National Museum of Korea), a palace you missed, or a day trip to Suwon's fortress.
If you add a Day 4
A DMZ tour (book ahead, passport required) or the folk village and painters' market at Heyri.