You may transport unloaded firearms in a locked hard-sided container as checked baggage only. Declare the firearm and/or ammunition to the airline when checking your bag at the ticket counterI've written about this before, but it has been a long time. Rules may have changed by now, so the above link is effective as of this date.
If you are travelling by air, there are some strict regulations which you must follow considering how your firearms and ammunition must be stowed.
Firearms must be in locked, hard-side cases. The plastic case your gun came in does not meet standards.
Ammunition must be also in a locked, hardside case. Only shotgun ammunition may be stored in the same case as the gun. (As if that's going to be big enough!)
Ammunition must be in a solid container, not cardboard.
Note: if you're going to a match or on a hunt, it might be easier to ship the ammunition ahead to your destination as a separate effort. It's probably a lot cheaper, too, considering the cost of shipping it as 'luggage' considering the weight allowances.
I once flew to New York from Oregon to attend a match. When I checked the baggage, I discovered that the nice lady at the check-in counter was not aware of the regulations. I tried to demonstrate that my pistols (in checked baggage) were unloaded, and it freaked her out. She had no idea how to determine that the guns were unloaded, and when I took the locked hard-case out of the luggage and opened it up, she was frightened. She had never actually touched a gun before, so I racked the slide on each pistol to show that the gun was unloaded.