I'm actually a current student from Omaha. The city is really what you make of it and it really has a small town feel despite it being a medium sized city (it's no Houston or Dallas, but you won't get bored). Here's the no BS run down of what Omaha has to offer:
The downtown area can best be described as downtown Austin scaled down about 20x. The Old Market district offers bars and restaurants that could be described as "hipster" and a lot of the shops offer unique items and clothing you will never find in a general mall. You're also probably aware of the Henry Doorly Zoo and Aquarium which is just south of the downtown area. If you have a family, I'd definitely recommend buying season passes with all the new additions they're adding. It's hands down the best zoo in the country (some may debate San Diego is better, but I'm biased of course).
If you're looking for cultural areas, Omaha really has it all. If you go south down 13th street, you'll hit our Little Prague area that has so many different foods and cuisines from Eastern Europe. If you continue down that way you'll come across Little Mexico, and when I say Little Mexico, I mean almost everything is in Spanish. I've found some of the best Mexican food down this way (if you're looking for recommendations, check out Sam Leon's on 20th Street; it looks sketchy, but it's so good). If you're not already aware, Omaha is very racially segregated. Generally speaking, South Omaha (East of 72nd Street and South of Dodge Street) is predominately Hispanic, North Omaha (East of 72nd Street, North of Dodge Street) is predominately African American, and West Omaha (Everything West of 72nd Street) is White suburbia).
If you're looking for a more intimate night out and don't want to deal with parking downtown, I'd recommend checking out downtown Dundee, which is about 10 minutes northwest of downtown. It has plenty of bars and small restaurants and it's surrounded by pretty well established neighborhoods that have been gentrified over the years (to put it in perspective, Warren Buffet lives in this area).
Shopping wise, there is Westroads Mall that has all of your basic needs: Von Maur (essentially a Nordstroms), Younkers (essentially a Dillards) and everything in between. Out west there is Oak View Mall (much smaller, but much less crowded). There are two outdoor malls, one in Papillion and one way out west (about 25 minutes from downtown). The one out west has a lot of higher end brands (Coach, Apple Store, Lululemon, etc.) since this is a very affluent area as well. If you're looking for an outlet mall, you'll have to trek about 40 mins from downtown in the city of Gretna. They just recently built this and has all major companies at outlet prices (Nike, Brooks Brothers, Polo, etc.)
Really Omaha has everything you really need. If you don't venture out, you'll hate it. Winter's are long and miserable, and summer's are blistering and humid (believe it or not, Omaha in July feels like Houston). The roads are noticeably crap because of the constant freezing and thawing and refreezing of the roads, so just prepare for a lot of chuck holes and on a more depressing note, I will mention taxes are among some of the highest in the country so prepare to weep when that time comes.
I love Omaha and I wish it didn't get so much grief because it's in Nebraska. I'm sure you'll love it just as much as I do. You're more than welcome to PM me if you have any other questions!