I would stay in Myeongdong, it’s the central shopping and dining district. Every night there’s a huge street food market and it’s one of the coolest things I’ve ever seen. Here are my suggestions for things to do:

Gyeongbokgung Palace (they have free guided tours in English)
Bukchon Hanok Village
Namsan Seoul Tower
Cheonggyecheon stream
Lotte World or Everland Theme Park
Coex or Goto Underground Malls
DMZ Tour (it takes all day but it’s incredible and the closest you’ll ever get to North Korea)
Gwangjang Market (featured on Netflix Street Eats)
Namdaemun Market
Korean Spa (jimjilbang)

The food is fantastic, especially the Korean fried chicken, Korean BBQ, bulgogi, and galbi.