Nebraska requires a CCW permit to carry within the state. Some states have reciprocity and information about what states have reciprocity are on the Nebraska State Patrol's webpage as they manage the CCW stuff.
Purchasing handguns (or blank AR15 lowers) requires a Handgun Purchasers Permit. I believe this is statute limited to $5, might not be, I can't remember. I know where I'm at, it's $5. Can be obtained at the local Sheriff's department or police department wherever you are going. Usually takes 1-3 days to get but is just a background check. Long arms don't require a Handgun Purchasers Permit, but it goes a lot quicker if you have one. Last time I purchased a long gun before becoming a LEO, the guy behind the counter at Cabela's groaned that I didn't have one, but the sale went fine without it.
Only other firearm restrictions in the state that I'm aware of pertaining to purchase/ownership, etc, is that Omaha has a registration for handguns. Lincoln shouldn't, and the rest of the state doesn't.
There are some laws surrounding carrying of firearms in the vehicle if they are loaded but a CCW permit pretty much eliminates all but the carrying of loaded shotguns in the cab of a vehicle.
Welcome to the Good Life State... I hope that wherever you are going in state, you are aware that come college football season, Lincoln's downtown area and most of the interstate headed to Lincoln (or out of after the game) are going to be PACKED. This state is CRAZY about the Cornhuskers, doubly so now that Scott Frost has come home. The vast majority of LEO's are pretty open minded about firearms as the state is very pro 2A... the State Patrol, Omaha PD and Lincoln PD are probably going to be the three LEO organizations to care most about the minutia about the firearms laws in vehicles, etc... the rest of us are pretty laid back about it!