There are over 63 miles of marked hiking trails in Custer State Park [1] ranging in difficulty from wheelchair accessible to strenuous. Hiking off of designated trails is allowed and, as a matter of fact, hiking is allowed anywhere in the park. Trail maps are available at the visitors centers [2].
Carry water with you on any hikes in the park, as water is not available along the trails. Camping is allowed only in the campgrounds [3] and open fires are not allowed along the trails, except in designated picnic areas.
Remember that afternoon thundershowers are not uncommon in the summer, so bring light rain gear. Watch where you step, as prairie rattlesnakes live in the Black Hills [4]. They are not aggressive snakes but will strike if threatened.
Finally, the only predator in the Black Hills is the mountain lion. They are shy of people and have never attacked anyone, but if you do see a lion, do not run. Maintain eye contact, make yourself as large as possible, and throw rocks and sticks.
Keep children close. Do not allow them to run too far ahead or lag too far behind the group.
Links:
[1] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park
[2] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/visitors-centers
[3] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/campgrounds
[4] http://www.moon.com/destinations/mt-rushmore-the-black-hills/discover-mount-rushmore-the-black-hills
[5] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/outdoor-recreation/hiking/harney-peak
[6] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/outdoor-recreation/hiking/sylvan-lake-shore-trail
[7] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/outdoor-recreation/hiking/sunday-gulch-trail
[8] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/outdoor-recreation/hiking/lover-s-leap-trail
[9] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park/outdoor-recreation/hiking/creekside-trail