I've always enjoyed this particular area of Korea because of Jaein Falls (재인 폭포), which is where this trail starts. It is about a 6.5km hike and changes around 400m in elevation, so it is a good workout and has some great views of the countryside.
Just a warning if you are not familiar with Korean trails, there are going to be someplaces that you will be wondering, "is this really the trail" as you try to climb up or down a almost verticle 10ft drop. I may be exaggerating a little, but here's a picture for an example.