Some of my Omaha favs:
Dixie Quicks for lunch. Awesome food, I have something different every time and it is always great. Definately call ahead for reservations.
M's Pub for late-night dining and drinks. Great atmosphere and very good food. Try the lavosh.
Mark's Bistro is also very good. I have only eaten there once, and enjoyed it very much. I had a wonderful carrot curry ginger soup. Their big seller is homemade mac n cheese with two different chef's embellishments that change daily.
Espana for decent tapas and sangria. This place is a bit of a hit-or-miss, but the hits are VERY good.
Sushi Japan for sushi. Very good selection and delicious.
Vincenzo's is pretty good Italian, and I am addicted to their soups.
Wheatfields has the absolute best breakfast in town.
Shucks has surprisingly great seafood. Seriously the best and freshest seafood I have had (except on the West Coast).
Guaca Maya has the tastiest and most authentic Mexican food in Omaha. Try the parrilladas or the Huanchinango a la Talla.
Jaipur has very tasty Indian food and they brew their own beer. Try the Jalepeno Ale.
The Bohemian Cafe has VERY heavy and delicious Czech food.
Whew. I know I probably forgot a few, but this should give you a bit to choose from. Also, what area are you staying in and do you have access to a car? That might factor in your selection as Omaha has a poor public transit system.
Try Stokes, southwest food with some french twists, they have more than one location but the one in the old maket is very good and has a great patio(12th and howard). Dixie Quicks on Levenworth Street menue always changes, you can bring your own wine (no liquor served), Jams on 78th & Dodge street california/american, Le Voltaire, country french, 156th & Dodge, Boiler Room in Old Market between 11th & 12th on Jones street, french and on the higher end as far as price goes. If you want a good steak try the Drover, old school steak house (dark interior) get the whiskey steak.