I've sent out about 10 sets of wheels around the country over the past 6 or so years.
Absolute cheapest way is Greyhound Parcel. If you go to a wheel shop and ask them for some boxes they usuallu have them around before the crush and recycle them. The wheels boxes come with foam and lip protectors moreoften then not.
You could ship a set of rims/tires across the country for about 120 dollars like this. The only hangup is you have to drop them off at a Greyhound terminal and the buyer has to pick them up... not to mention its slow. Its also only possible to get 300 insurance per box. So if the wheels are expensive I would absolutely not use this method.
FedEx/UPS will ship your wheels/tires across the country for about 180ish-200 insured packed in your boxes(sourced the same way).
If you want to but packing boxes & material, have them packed and then shipped. Your looking at probably 300-350 dollars.
The more work you do before shipment the less it will cost you.