My boyfriend and I love Kukje Gallery in the Samcheong neighbourhood of Seoul. Located just outside the Gyeonbokgung Palace walls, the gallery sits on a pleasant, leafy stretch of interesting glass and concrete buildings and is a great place to see well-curated shows of both Korean and international artists. You can learn more about the space here.
Of course, my favourite part of Kukje is the Cafe, located on the first floor of the gallery. It's a great place to sit and have a pot of tea and a piece (or two) of cake, which comes served on very pretty plates. They also make the best hamburger I've tried thus far in Korea (I think the secret is the slice of grilled eggplant). The decor is very airy and bright, and the huge windows are great for people watching.
The Gallery has two separate spaces, the second located just behind the first. Here, their large library of in-house publications are for sale. The printing presses in Korea are absurdly good - the books are not only beautifully made but so reasonably priced, making them great additions to any library. I bought the Kwang-Ho Lee book on my last visit and would love to have so many more.