One of the ultimate places to go to as a couple in Korea is definitely Namsan Tower. We took a bus there from Insadong (if I remember correctly) that took us up the hill, but not all the way. There was still a stretch of steep road left to climb, but it was totally worth it for the view alone. Once we reached the top and snapped some pictures, we went to the little plateau that that love locks all over. There’s a little shop at the bottom that sells padlocks and pens, so you can write your name and the name of your loved one and lock it onto the fence. We then went into the actual N tower, where we were taken into a very trippy elevator with some cool projections that took us all the way to the top. Now, the view from there was truly spectacular and you could practically see entire Seoul in 360. We also took some funny montage pictures (as you do), then went into the restaurant at the bottom for post-sightseeing treats.
If you can, bring your own pen and padlock, because the ones sold out there are terribly overpriced!