One of the top zoos in the country, and a great way to spend some time for both kids and adults. Our favorites are the aquarium and the gorilla enclosure.
The Riverfront area is a large park starting in downtown Omaha and extending to the Missouri River - a great place to hang out in nice weather. It includes a couple of large playgrounds and a water play area for the kids (one playground is at 201 Riverfront Dr, and the other two areas are roughly between 10th and 13th Street on Farnam). Downtown has a ton of great restaurants/bars, and some interesting little stores.
The Joslyn was redesigned over the past few years, and just recently reopened. Admission is now free. There is also a cool exterior sculpture garden area.
Our favorite place to walk/hike in the area, located about 20 miles north of downtown Omaha in the Loess Hills. Includes an observation tower with great views of the surrounding area.