The old palaces in Seoul are really worth seeing especially a place called 창덕궁 and the secret garden there (후원). It's one of the most beautiful places I've ever been to, so I highly recommend it.
Also, it might be nice to check out 경주. I took my French friend to a few places around Korea, and it was his favorite place because there was a lot of historical sites around the city and overall, it feels more like traditional Korea than Seoul (Seoul is interesting to walk around if you want to see the modern history of how Korea developed). 불국사 which is a big temple place near the city is also very beautiful and 석굴암 that's near there is also worth visiting.
And well.. It would never end if I went on, so I'll just list the two that's on my top list :)