Brown County is a good spot. I think pretty much all of the state parks offer primitive camping areas if that's what you have in mind, and there's generally a town close by in case it gets too rugged for the young-uns.
Your very close to Potato Creek state park. They have everything a camper needs............cabins..fishing...hiking...rent bicycles. You can pitch a tent or bring a camper too
I went to Turkey Run for the first time last year. Its a bit of a drive, but the campground was pretty nice and it has some great trails. I highly recommend it.
Since its your first time, I'd agree with Brown County. The primitive campground is nice and also has some good trails.
The backyard is a excellent place to start, and is nothing to be embarrassed about.
It's much better than driving out to a state park, unpacking everything, and finding out you absolutely hate camping, or that your significant other(s) does. Plus you can work all the big bugs in your camping plan before you have to scramble in the woods.
Tink: The Charles C. Deam widerness. Part of Hoosier National Forest, just South of Lake Monroe. You can park at Hickory Ridge Fire tower and there are primitive campsites right there. There is also, off the beaten path camping any place in the whole forest(all the way to the lake.) IMO one of the best places in the state to camp. Barred owls at 4:00a.m. Badgers at daylight.
If you are from So. Bend... Try Tippecanoe St. Park close to Winamac IN. They have a great staff and als on the western side of the park is a public outdoor shooting range.. Might get a taste of the survival life and a little weapons training in one stop!!!
Any state park would be a good place to start. Pick one close to home so if there are issues you can take care of it easily. My personal favorite is Whitewater SP south of Richmond.
Pokagon State Park is just a quick trip across the toll road, and it offers rustic camping as well as a beach on the north side of Lake James. I'm not sure when the stables close for the year, but they used to have trail rides there also.