Great outdoors close to the city!
Hitchcock Nature Center is north of Crescent, IA and just let outside Omaha, NE. A short drive from the city and you are in the Loess Hills that were explored by Lewis and Clark on their trek west. The is a $3 day use pass or a $20 yearly pass available.
This place has a lot to offer anyone. They have Nature education events, archery range, a handicapped accessible boardwalk trail, hiking trails, camping, etc. There are miles of trail available with backcountry camping available for $10 per night. RV camping is also available for $15 per night with electric, showers/bathrooms, picnic tables, fire rings, etc. They also offer a couple cabins for rent.
The hiking is fantastic and offers something for every skill level. There are absolutely gorgeous views of the Missouri River valley below. All trails are marked and the a multiple different trailheads, many which start around the visitors center/tower Overlook. During the winter they even over snowshoeing events. It is really a nice place to get outdoors anytime of the year.