Downtown is mostly business people/professionals during the day and young people/bar hoppers at night. Amazing zoo, huge venue for the more well-known bands that roll through.
The midtown area and Dundee are pretty diverse with lots of different age ranges and cultures. Very walkable. Also has a few bars and restaurants but they are more.....relaxed is the best way I could put it. You'd go there to have a few beers with friends, whereas you'd go downtown to party til 3am, get completely wasted and dance.
South end of town is a bit more industrial with a large Hispanic population. Great Mexican food and fun little shops. As you go further south into Bellevue, there's the air force base. Once you go far enough to hit Sarpy county, you better be on your best behavior, as the cops are well known for being arseholes.
North Omaha is the oldest part of town. Benson is a fun little neighborhood with a couple good small venues for live music. Farther northeast is filled with predominantly black neighborhoods. Fantastic BBQ and while it used to be pretty run-down, they've made some improvements to local businesses and I think the area is getting better. Still, there are gas stations where you cant go inside after dark and have to pass your money through the window with bars.
West Omaha is full of cookie cutter newer homes as well as more affluent neighborhoods. Target, Walgreens, or Applebees/chain restaurant on every corner. Very safe, but not much to do.
For an incoming student, I would highly recommend the Dundee or
midtown areas.