I will leave pet-friendly to you to find out. Just do some kind of search on motels in areas that you intend to stop that allow pets.
My idea of a nice drive is to avoid cities as much as possible. Particularly, NY and Philadelphia. And also avoiding I-95 where possible as it seems always to be congested.
So I would start out from Mashpee
and take I-495 N, I-90 W, I-84 and I-81 S around NYC via Scranton
(first stop) to Hershey
Hershey and the Pennsylvania Dutch area has some interesting places.(second stop) Then cross down to Baltimore
and go over onto the Eastern Shore (third stop) to Cape Charles
and take the Bridge Tunnel to Norfolk
(third stop) Then just stay on US Route 17 to the Dismal Swamp Canal Welcome Center
and on to Wilmington
(fourth stop) Myrtle Beach Charleston
(fifth stop) and Savannah
From Savannah you can take I-95 and then I-75 over to Sarasota