This!

We LOVE the HD Zoo! There is a ton to do! The aquarium is fun, but you will definitely have plenty to fill your day...or two days.
The desert dome, creatures of the night, rainforest, skyfari, feeding the koi while the spider monkeys try to steal the food...I could go on and on. We went just as adults the first time and still spent the entire day. People from KC drive 3 hours for a one day trip to go to HD Zoo..and we HAVE a zoo!
You can also take your food in and have a picnic to save money. We packed an entire lunch and they didn't say a word. I can't even remember if they checked our bags or not.
We get there at opening. When you are facing the entrance to the zoo, go to the bottom left of the parking lot. There is another entrance down there and people fail to realize it. That part of the zoo is empty at opening because so many people go through the main entrance.
There is a place called King Kong's right outside the zoo. I love the gyro's there and get them as we leave everytime. They are about $7-8, but worth every penny! Anytime I go to NE, we eat at King Kong's. It's a fast food place with burgers/gyros and plates of fries. Definitely very unhealthy! And delish!
We usually stay with family friends while we are there...or at the Mahoney State Park campground near Gretna. The Zoo's Wildlife Reserve is out there. It's a drive through reserve and then there is a place to stop and walk to see wolves and bears. Where you stop, there are zoo workers (??) there that you can talk to about the animals. There were bird exhibits and crafts when we were there and we learned so much. Our kiddos LOVED the Wildlife Reserve, since the buffalo and elk were coming up to our windows. My DH was scared and kept rolling up his windows.
Have fun! Omaha actually has a lot to offer!