My husband's family lives in Omaha and there are places we hit up every time we visit:
Big Fred's Pizza is one of those hole-in-the-wall places that, as soon as you walk in the door, you *know* the pizza is going to be good.
Gorat's is a great steakhouse when other people are paying :)
The Garden Cafe has huge portions (seriously, unless you are sharing with four people go with the "junior" size of everything [you'll still have leftovers]) and I highly recommend the potato casseroles. They also have great breakfast stuff and the bakery stuff is great.