A few friends and I went to Seoraksan (설악산) in October to catch the leaves changing color.
There are two things I have to recommend:
First, hike to Ulsanbawi (울산바위), a rock formation in the Seoraksan national park. The shape of Ulsanbawi is unique in the area. To reach the rocks you need to follow a hiking path and climb over 800 steps (or 888 steps according to locals). On the way there, there are two temples and a spherical rock (Heundeulbawi, 흔들바위) which is located on top of a larger rock.
According to the legend, Ulsanbawi comes from the city of Ulsan in the south east of Korea. As Kumgangsan (금강산) was built, Ulsanbawi walked to the north, as rock formations used to do. Unfortunately, Ulsanbawi arrived too late and there was no more room. Ulsanbawi headed back home in shame. One evening the rock went to sleep in the Seorak area. Ulsanbawi thought it was so beautiful, it decided to stay for good.
The second thing you should do is take the cable car up. Buy tickets for the ride right when you enter the park. During peak time, tickets do sell out. The ride is fun, and the view up there is incredible.
To get to Seoraksan, I recommend taking a bus from Seoul Express Bus Terminal to Sokcho (속초) Express Bus Terminal. I recommend buying bus tickets a few days in advance. The bus ride is around two and a half hours, and tickets cost 17,000₩.
Once in Sokcho, cross the street and take Bus 7 or 7-1. Get off at Seorak-dong (설악동, the last stop). For accommodations, there are hotels near the bus terminal that don’t require prior reservations (~40,000₩ a night for two people). Another option is renting a pension where a bunch of people can sleep on the floor (~60,000₩ a night for four people), but you’ll need a Korean speaker to reserve it over the phone.
Seoraksan is the most beautiful place I have been to in Korea, and I definitely see myself returning there.