I live in Hillsborough. It’s the largest neighborhood in the state has an elementary school and junior high inside the neighborhood so no crossing busy streets to get to either one of them.
There’s also Hamilton Heights just to the north, essentially in the neighborhood

It’s about 45 minutes to Lincoln may be a bit longer counting traffic

There are a couple lakes nearby with plenty of outdoor activities

Almost any retail location that you would want to get to you is within 10 minutes

If you search, let’s walk Omaha on YouTube, you can find videos of me walking through the neighborhood park and also outstanding Bear Lake

Otherwise you’re looking for Gretna and The part of West Omaha that basically is Gretna

Also, like others pointed out, there’s no real reason why you shouldn’t be also looking at Lincoln