Answer:
I recommend the Henry Doorly Zoo - an AMAZING Zoo http://www.omahazoo.com
I have a season ticket and visit there regularly with my family. Go here to see some pictures and video of a small part of it taken during a recent visit http://picasaweb.google.com/ian.barker/OmahaZooApril2010# it also has links to previous visits.
Also, plenty of nice restaurants in "The Old Market" area of Omaha.
You should also consider The Joslyn Museum - http://www.joslyn.org - lots of fantastic art, paintings, sculptures and so on including some by Rodin, Matisse, Picasso etc etc. Really really worth a visit.