I'm in Omaha, the Zoo here is only surpassed by San Diego's, and not by much. St. Louis Zoo cannot hold a candle to Omaha's.
I have not been to the UP museum across the river in CB, so I cannot comment on it.

Buffy's posting is pretty accurate, but Caniglia's Steakhouse is closed for good, a victim of casino food (or so they say). The best Italian easily is a place called LoSole Mio near 32nd and Oak street, it can be a 2-3 hour wait to get in on weekends. There are a lot of 'authentic' Mexican restaurants here, but the word 'authentic' is open to interpretation. Personally I like a place called 'La Mesa', there are 2 or 3 of them in town, followed by Fernando's and La Fonda.

We are also folks that used to go to KC to shop, but anymore, just about everything we went to the Plaza for or Overland Park for is now in Omaha, but we still don't have a danged Cheesecake Factory yet !

Airports: LAX is a snap to get thru (security-wise) compared to Omaha, so far Omaha is the worst, slowest one I've ever been through, but I have only been thru LAX CHI LV and PHX.

Tell me where you are staying and I will tell you where to eat that's close (anf good).

Nebraska itself: I'd go to the sand hills near Valentine or Chadron, but I am easily satisfied: 60 miles south of Valentine you are literally nowhere, in a telescoping black hole that is one of the darkest areas in the entire USA.
The only place as dark are some spots in Nevada.
It's AWESOME if you spend a night there, roll out a blanket, close your eyes for 5 minutes, the open them. The sky looks like it's on fire with all the stars you can see with no light pollution. Folks in cities have no idea what it's like.
There are national telescoping 'parties' there in the summer where hundreds of people camp out just to see stars you cannot normally see anywhere else.