Gunnar Caya
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Mudeungsan National Park is the closest park to Gwangju, which is pretty well connected to the rest of Korea via train and bus. It is also one of the easiest parks to reach using public transport, as the 1187 and 9 bus take you to the two main respective entrances to the park, and they both run pretty frequently.
As far as the hiking itself goes, having hiked in all but two of the national parks in Korea, I'd say it is in somewhere in the bottom half in terms of the views, quality of hiking, variety of routes, etc. The route from Wonhyosa especially has very little to see besides thick tree canopy for most of the hike. The trail from Jeungsimsa has better views after you've hiked uphill for about an hour or so, but still pretty ordinary for a Korean National Park.
That said, it is still quite nice and much better than your average provincial park or random Korean mountain on the edge of town. The trails were well-maintained and well-marked for the most part when I visited in August 2024 (although they need delete Cheonwangbong and Inwangbong from the signs or say clearly that these are off limits  to the general public). Plus you will find some pretty unique rock volcanic formations here, which though not as impressive or varied as Juwangsan and Hallasan, are still quite a change from the typical Korean mountain.
If you're looking for a national park (the mountain variety of course) that is easy to get to from a major city with relatively little hassle, this one is probably the easiest to get to besides Bukhansan and perhaps Gyeryongsan. As a result, I'd definitely recommend it.