One of the prettiest places to see in Seoul would have to be Biwon, or the Secret Garden of Changdeokgung Palace. Accessible only via guided tour, this is one tour you don't want to miss. The English tours occur twice a day, at 11:30 and 2:30, and cost 3,000 won. Entrance to the main palace costs 5,000, and you can visit that at your own leisure during open hours.
I've taken the Biwon tour twice in my time here in Korea, and both times I have equally enjoyed its beauty. The best times to go are in the spring when everything is fresh and in bloom or in the fall, when the trees explode into brilliant colors.