Here are some that I have liked to frequent. Many are around midtown since that's where I worked:

Breakfast:
Wheatfields (Good anytime)
Vidlaks
Farmhouse Cafe
Joe's Egg

Lunch:
Wingstop (Franchise but has great variety and better than Buffalo WW in my opinion)
Charleston's
Johnnie's Italian Steakhouse (Lunch specials. On Thursday they have a strip, salad, and drink for @ $10)
PepperJax Grill (Philly style sandwiches)
Spezia
HuHot Mongolian Grill

Dinner:
Charlie's on the Lake
Flatiron
Sullivan's
Upstream Brewing (Try the root beer)
Plank Seafood
Bonefish Grill
Gorat's (Bill Gates and Warren Buffet have been seen here dining together. Food is ok and it is an older place but it is traditional Omaha)
Mahogany Steakhouse (very high end. Great if on company's dime.)
Paxton (Another High end steakhouse)

Omaha is a very casual town. You will find casual dress in even the most upscale restaurants.