How much you pay to ship a quilt depends on how much the quilt is worth to you.
If you can easily replace the quilt, then send it in one of the flat rate priority boxes. I wouldn't buy the insurance or the delivery confirmation - they'll only pay out on the insurance if you have the quilt appraised, and my experience with the delivery confirmation is that they'll post the confirmation long after the box has been received, although they may have improved the service.
If it's a quilt that can't be replaced, then investigate FedEx or UPS. Both provide online tracking so you can watch it go across the country. Have the quilt appraised and insure it. Registered mail is also a good option - it's slower, but they inspect the package at every stop.
Janet