Two more places that I recommend first-time visitors to Korea to experience are "N Seoul Tower" (N서울터워) and "Myeongdong" (명동).
N Seoul Tower is a great spot for a date, since you're up high above the rest of the city and it has a nice atmosphere. But it's also fun for visiting to spend a few hours relaxing.
Myeongdong is a modern shopping center, and you can find everything from souvenirs to makeup supplies to ice cream.
Check out the video to enjoy a quick tour of these two places.