Omaha has Lyft and Uber. I would suggest using them as the local cab company (there is only one with multiple names and phone numbers) sucks.
Getting around when you're drinking is an issue. State law in Nebraska is that you can't flag down a cab. You have to call one. Our public transportation is dismal at best. Here are some numbers for some cab companies.
You must provide a proper address for the cab companies here. They won't just come to a random corner. Ask a bartender. They also get overwhelmed on St. Patricks/New Years. Plan accordingly.
Happy Cab/Cornhusker Cab
(402) 339-6300
City Taxi
402-933-8700
I have never figured out Omaha Public Transit. You have to wave down the busdrivers, as they think that you're hanging around under a bus stop sign for fun. It's easiest to do public transit map search on Google Maps. However, busses can be 15 minutes late or EARLY... which is frustrating as hell.