Go Big Red! I used to live in Nebraska... moved to Nevada to go 4 wheeling!
There isn't much opportunity for off road recreation in Nebraska, but for camping and some back country feel, you could head for the Nebraska National Forest (the only hand planted forest in the USA!) near Halsey. It's a couple hundred miles from Omaha, but there are dirt roads to explore, lots of trees, and some fairly nice camping locations.
I've read there are some 4x4 opportunities along the shoreline of Lake McConaughey, but you are getting pretty far from home there.