This gorgeous park can be a hard day of hiking (if you park at the lower free parking lot) or an easy day out with the kids on smooth paths perfect for strollers (if you park at the top in the paid parking area & entrance fee. Worth it. Maybe 3,000 per person). If you are a family with mixed levels of activity, it's the answer you have been looking for. You can drop your avid hikers off at the bottom of the mountain, then drive to the top to lounge about in the forest parks and rec areas while they huff it up to meet you.
In addition to hiking, there is a swimming pool (open only during "swim" season starting in July), a nice zipline, obstacle course, play area, rental cabins, museum, cactus arboretum, water features, basketball court, badminton court, etc. You get the picture, plenty of stuff to do and a peaceful place to do it in.
This is another place we have driven past for 5 years and never stopped until now, we regret it. This is the nicest place for hiking near Pyeongtaek by far. Incredibly well maintained and refreshing, great views at the top.