I'd either just take Acela or egional pvd to Penn station. If you want to end up at grand central, I personally would take shore line east from old saybrook to New Haven and then transfer to metro north to continue to grand central. It's very straightforward but may take a little longer. Flights from pvd to NYC are kind of cheap but parking is not free, the airport is nowhere near Providence, (though you can get there by rail). And NYC airports are not really in NYC either. The flight itself is like 40 minutes though