For burgers, forget everything else and go to Blanc Burger at the Midtown crossing. Another casual, but great place at Midtown is Ingredient.
For fancy, I recommend V-Mertz in the Old Market, and Grey Plume (I think someone recommended this one too) at the Midtown Crossing. Amazing seasonal menus and local ingredients. Fantastic chefs.
For unique Omaha eats, and a great sandwich - go to Shuck's Fish House and Oyster bar for an oyster poboy. Also, Amsterdam Falafel in dundee has what I consdier to be the best sandwich in Omaha. It's a dive, ran by two twenty-something hippies, who make amazing falafel. They have like three things on the menu - falafel, kabobs, and curry fries - amazing!
If you are in town, you may consider spending some time downtown in the Old Market. The area is full of shops, great for a quick walk and people-watching, and of course, restaurants, including what many consider to be the best steakhouse in Omaha - Omaha Prime: http://www.omaha-prime.com/.
I also recomend V Mertz: http://www.vmertz.com/. This is not a steakhouse, but is an upscale restaurant with a seasonal menu. The reason I recommend it is because they have one entree that is always on the menu, and for good reason, the V Mertz Pepper Steak. Amazing. Again, it's not a steakhouse, so you won't have an array of steaks to choose from, but it's great.
Another good steakhouse in the Old Market area is Sullivans: http://www.sullivansteakhouse.com/. Great service, and good steaks. they also have some good seafood, but who wants to eat seafood in Omaha? It is a chain, and I tend to avoid chains, but thought I would include it as an option.
I focused mainly on the Old Market area of Omaha, but I'm sure you will get other recommendations that will be helpful for you, inlcuding Flemings, Gorats, and 801 Chophouse at the Paxton - all good options. Enjoy your stay!