There was just a discussion on this on another thread. If size is right - priority mail flat rate boxes from the post office are good choices - they can be tracked along the way and are a reasonable price. If large - sometimes FedEx is a better choice - again they can be tracked and can be cheaper than the post office. I am not a fan of UPS.
You can go on line to USPS, FedEx and UPS and put in box weight and dimensions and get prices. There is something that not everyone is familiar with call "DIM weight"....but putting in weight and dimensions on these sights will cover that contingency.
All can be insured, all can be tracked.
PS>........i ALWAYS put mine in a clean plastic garbage bag before putting it in the box....that way, if for some reason the box gets wet, it doesn't get to the quilt.
For a ginormous quilt - i've put it in a space bag and sucked the air out, then boxed using nylon strapping tape (just in case the suction doesn't hold and it tries to expand.
Good luck.