I'll second (or third or fourth...) the zoo. another reason to see it is the desert dome. Its something like the larges free standing geodesic dome in the US (or something like that).

Also I have heard that the SAC museum is intersting to see. It is about 30-40 minutes west of Omaha. Also, if its your thing (its not mine, so I can't tell you if its any good) but there is a pretty large casino in Council Bluff, just head East on 80, you can't miss it.

As far as food goes, I really like pepperjax. Its cheesesteaks, but decent ones are hard to find in the midwest. You also can't go wrong with beef. There are a number of good steakhouses in town.

Finally, the downtown area (I think its called the haymarket area) is a neat area to go for resturants or bars. There's a pretty good brewpub down there, just down the street from the courtyard (Marriott). Supposedly there's a great hole-in-the-wall steak place right across the street from there, too. Sorry I don't have names, I usually just get there by "feel."