I have a couple more to add.
Geoje Do (Island) on the southern coast. When you get into the national park there is very little development but you can still find a few places to stay. Totally beautiful!
Also on the south coast near Yeosu is Hyang Gil Am (Temple). It sits high on a cliff overlooking the sea. You have to walk through narrow passages of rock to get around the temple grounds.
Im a big fan of Hahoe Folk Village in Andong
Though its always busy, Songnisan is my favorite mountain