JFK, Newark (EWR), and LGA are the closest international airports to me (other than SWF which is a joke). I would consider flying into Newark and taking Amtrak directly from the Airport to Providence, RI. The trip is about 4 hours (yeah, US trains are slow). I have made this trip twice and it's not horrible. Upgrade to Business Class since it's cheap (about $20 extra) and you get 110V 3-prong power outlets at each bank of 2 seats. The seats are also bigger and you get a pass for free beverages in the cafe car. One time, when I was in Business Class, the conductor was giving people tips on finding a seat in the coach cars at every stop. I was quite happy to be in a half-empty car with a whole row to myself.

You can catch the train from JFK and LGA also, but you will first have to take a bus to Penn Station. That is lame and will probably take over 1 hour depending on traffic.

Once you get to Providence, I guess you would rent a car somewhere. The station is well-staffed and pretty modern (compared to other Amtrak stations). It's in the middle of the city, so a car rental agency should be around somewhere. Or, maybe you could call a taxi service since getting anywhere within the state of Rhode Island isn't very far away.