@SusieGarriott , thanks for asking! There are so many wonderful state parks for day hikes but for backpacking destinations within 2.5 hours of Indianapolis, our local employees recommend Shades State Park or Morgan Monroe State Forest. If you're willing to drive just a little bit farther, check out Daniel Boone National Forest (Red River Gorge) or Shawnee National Forest. Happy backpacking!
@SusieGarriott I love backpacking in Indiana Dunes National Park. There are no primitative sites, but you can easily plan an out and back or one way route utilizing the Dunes State Park and Dunes National Park campgrounds.
You're about 3 1/2 - 4 hours from Shawnee Nat'l Forest, but there are some beautiful places there, lots of places to make loops by combining the River to River trail and other NFS trails, especially around the Garden of the Gods area.
Deam Wilderness and Hoosier National Forest are the closest and great for quick weekend getaways, however if you want more scenery and are ok with a longer drive (4-5h) definitely check out Shawnee National Forest or Red River Gorge.