When shipping nesting bowls, I typically wrap each bowl in tissue paper and bubble wrap, then next the bowls together and securely bubble wrap the entire set together. I then wrap this package in kraft paper or newspaper.
Make sure you still have several inches on each side to cushion the package in the box. In my experience, a USPS Flat Rate Large Box is NOT big enough, but if shipping parcel post the cost is about the same domestically - ~$15 (and be sure to insure).
I want to edit to add: Especially when shipping a rare or highly desired pattern (or multiple pieces), I double box. I do as I described above and then place that box inside a slightly larger box, with about an inch of cushion. In my experience, customers do not balk at the cost of shipping/packaging if it is a) realistic to cover the cost of materials and b) if it means their fragile item will get there safely and in pristine condition.
