When I'd sell a quilt on Ebay, I mailed parcel post. I always put them in garbage bag just in case the box got wet. It takes maybe a week to ten days for Parcel Post.
You can go to usps.com to check costs for that, and also for Priority rate. You can get the free Priority boxes from the post office, or if you have time, they'll mail them to you. Priority is a lot quicker, but Parcel Post is cheaper.
I agree, always get a tracking number, and use lots of packing tape to make sure the package can't come open.
If you need a box for it, you might want to ask at an auto body shop. They always have good clean boxes.
:-)