If you have the opportunity, Changdeokgung should also be on your must see list. It's on UNESCO's World Cultural Heritage List. The palace has been extensively restored and it's one of the greener parts in the city. I found it a lot nicer than Kyongbokgung.
Other things to do/see: Dongdaemun shopping area; Daehangno, a great place to hang with lots of college students; Insa-dong (most cultural street in Seoul--the Starbucks sign is in Hangeul); Apgujeong, Gangnam (where the rich and young hangout); DMZ (there are tours up to the demilitarized zone, but it's only for non-Korean citizens); there's even more to do.
Other things to do/see: Dongdaemun shopping area; Daehangno, a great place to hang with lots of college students; Insa-dong (most cultural street in Seoul--the Starbucks sign is in Hangeul); Apgujeong, Gangnam (where the rich and young hangout); DMZ (there are tours up to the demilitarized zone, but it's only for non-Korean citizens); there's even more to do.