Has been several years since I lived in Omaha but here are a few of my favorites:
Indian Creek (27 holes) -
https://www.golfatindiancreek.com/Quarry Oaks -
https://quarryoaks.com/This is a bit west of Omaha, headed toward Lincoln and just off the interstate. Some good views along the Platte River.
Bay Hills (Plattsmouth) -
http://www.bayhillsgolfclub.com/course/Has been a really long time since I played out here. Used to be a good, challenging course. You may want to check it out before making the drive south of Omaha.
I am sure there are several other options to choose from. These are three I have played. IMO, you have to play Quarry Oaks if you in Omaha.