Definately the zoo. Its one of the best out there. Depending on your tastes, there are some casinos in Council Bluffs (just over the river into Iowa). There is also the SAC (Strategic Air Command) musuem which is about 20 minutes or so west of Omaha. Haven't done either of those, but from people who like that sort of thing, I have heard that they are pretty good. Not really outdoor activities though
Also, FWIW, I wouls stay near Haymarket or downtown. Some of the other hotels in Omaha are starting to show thier age.
I will look up some of the other resturants that are good out there, and post them when I get the names.
AJ