I wrap our masks in tissue paper and then stuff the box (lightly) with crumpled newspaper so there is no shifting around. Also, be sure to use good quality boxes. (I do reuse boxes sometimes, but only if they are not beat up to the point of being "unsound.") We've never had problems with damage since I hit on our current method of packaging.
As for shipping, I prefer USPS to FedEx, etc., because of cost, as well as service. I alternate between First Class and Priority; both are affordable and I use Priority when I need it to get there just a touch faster, or when the package is too large to go First Class. I have had people request/use Overnight shipping, but it's pricey, so I only do it on request.
Good luck! :D