Jirisan National Park is one of South Korea's best places to hike. I've made a couple stabs at the hike from Hwaeomsa Temple near my home in Gurye to the Nogodan shelter. The scenery is beautiful, the trails surrounded by mixed forest of deciduous, conifer, and bamboo trees. It can be rocky in parts, and I recommend sturdier footwear than running shoes if you're going up for more than a day.
And I heartily recommend going on a multi-day hike here, if that's your thing. Maps (1:75,000, topographical, but only in Korean) are easily found at the bus terminal in Gurye and at the trailhead. Plan ahead, and make sure you're strong enough; my attempts up the trail left me wobbly-kneed at the halfway point.
Pronounced Hwa-om-sa, this is a beautiful temple set at the bottom of a valley with a rocky river with many little waterfalls and pools. The buildings are historically significant, and there is a well-written and informative pamphlet in English. There is also a nice hermitage uphill from the main temple. If you're going there, I recommend the shorter path along the trail to Nogodan, which is very pretty. My second time there I had a nice tough hike up the rocky trail (well past the hermitage) and finished off by sitting on one of the rocks by the restaurant and watching children play in the river.