You'll need either a NE Firearms Purchase Permit (Blue Card) or NE Concealed Handgun Permit to purchase a handgun in Nebraska, whether from a licensed dealer or an individual. Either one will work OK. However, only NE residents are eligible for these documents.
You could physically transport your personally-owned handgun from your home state to Nebraska/Omaha under federal law (18 USC 926a). Then that handgun would need to be immediately registered with the Omaha Police Department, first thing upon arrival.
You can rent a handgun at a commercial shooting range without a Blue Card, if the range management will so permit. (Many will not rent without a Blue Card, regardless.)
You can borrow a handgun for shooting so long as the legal owner is with you in immediate visual and aural contact.
Your first step is to clearly determine your legal state of residence, i.e., where you are registered to vote, your auto is licensed, driver's license is issued, place of work, mail is delivered, etc. You may actually be a Nebraska resident whether you realize it or not. You would benefit from a discussion with an attorney on that point.
Probably not a good idea to ever acquire (purchase, own, possess, receive in transfer) a handgun until all ducks are lined up in a nice, neat, clearly legal row.