I visited Leeum Museum of Art, one of South Korea's top private art museums, and admired the clay pottery and mother-of-pearl inlaid objects and jewelry, though the contemporary sections were closed during my visit.
I visited the Kyung-In Museum of Fine Art, which features different rooms with various artist exhibitions, and I loved its very pretty, traditional Korean style, feeling it was very much my vibe.
The Dongdaemun Design Plaza is a ginormous plaza full of different design things; its building is beautifully created and designed, making it something you can't miss while taxiing around the city.