Since I live here I thought I would take a shot at your question. 284 from 111 is considered the back entrance to the park and it will get you in but it is a bit longer around that way. If you're coming down 111, I might suggest picking up hwy 30 east in Spencer which leads you to the front entrance of Fall Creek Falls, especially if you're up for some heart thumping twisties. That section of 30 is pretty exciting to ride. Once you get into the park watch for signs that will direct you to the Inn (restaurant).
After you enter the park, the first intersection you come to is the nature center. Go straight thru that intersection into what looks like a prking lot. Keep going to the end of the lot and a road exits down the hill. That road merges back into the main park road at the top of the hill and will lead you to the camper check-in-station which sits in the middle of a 3 way intersection. Bear right at the camper check-in and stay on the that road until it come to a left-right split. Go left and the road dead ends at the Inn.
On a side note, if you come in on Hwy 30, you will pass by the entrance to the recently discovered Rumble Room Cave. It is the 2nd largest cavern ever discovered in the U.S. The Louisana Superdome would fit in the main chamber with room to spare. As soon as somebody figures out how to open it up to tourism....look out! There are pictures posted on the web if you search on "rumble room cave" if anybody wants to see them.
I should be at the first rest stop on I-24 in Tennessee between 7:30A & 7:45A. I believe that is at mile marker 93. From there I'll head to Nashville, take Briley parkway over to 40. I'll be glad to look for you at the rest stop.