Oh, sorry, my mistake. There are indeed MBTA trains from Providence on Saturdays.

The cab from PVD airport to downtown runs about $35 - $40...I have had to do this a few times in the last year when returning from business trips.

I cannot believe that I did not think of this first, since I have actually done this trip several times for business.
Peter Pan also serves Newport and there are daily busses into Boston from the bus terminal right in Newport. The schedule lists a daily bus at 2 PM and a daily bus at 7 PM. This would appear to be your easiest choice. The trip is about 1.5 hrs and would be a lot less hassel than the other choices

But that still leaves the OP needing a way to get from the airport to Newport. And they'll have luggage, so a rental car is almost essential for storage as well as for transportation.

I agree anonymous, but she now only needs the car to get from PVD to Newport.return it in Newport and jump on the bus!

EmilyC, it would be easiest if I can drop the car off in Newport. Let me look into it...At the worst case scenario, I will drive the car back to the train station and take the Peter Pan into Boston South Station.