The picture above is a good place to check out (Bukchon).

You can start at Insadong (1) which has a bunch of stuff to buy. Then walk north, up to Bukchon (2), see some nice old hanok houses, hit up a tea house, then head over to Changdeok palace (3) and then to Hyehwa (4)

That can be an entire day, or leave Hyehwa to another day by itself. It sometimes goes by DaeHangNo. That green spot to the west of 2 is the other palace, Gyeongbokgung but I think it’s not as interesting