The open plaza area in front of the Sejong Cultural Center was a frequent visiting spot for me when I lived in Seoul. In the summer months the cool water was a relaxing spot to wet your feet and watch children running through the musically choreographed fountains.
I would take the bus to get to this area but the best way by subway is to look for Gwanghwamun Station on the Purple Line, Exits 2 and 3 are best. Once you've gotten enough rest and are ready for more sightseeing walk down past this fountain and you will see Gwanghwamun (gate) and Gyeongbokgung (palace), both are must see attractions in Seoul.
At night time this fountain and plaza are more appealing as colorful lights adorn the grounds.