Definitely go to Seoraksan.
Also, House Hostel is arguably the best place in the city to stay. It's run by a remarkably friendly guy, whose name I forget (I think it might've been Mr. Yoo), and it's immaculately clean. Rooms were 20,000 for a single or 30,000 for a double. Big, comfy western-style rooms, too. Also there was free internet and free breakfast and coffee.
It's just a few minutes away from the bus station to Seoraksan as well as the bus out of the city. And close to one of the seafood markets (where the crab and sea urchin are phenomenal).