You can take the commuter rail directly to TF Green Airport at certain times of the day. Check your arrival to the schedule at www.mbta.com
Note that not all trains to Green Airport stop at Back Bay, for those trains you'll have to go to South Station, which would mean Orange Line to Red Line. Easier to take a cab.
To clarify, all trains to Providence whether Amtrak or commuter rail leave from South Station. There is no connection between North Station and South Station, so you have to make your way to South Station by subway (requires a transfer from Green to Red at Park) or by cab/Uber/Lyft. It actually can be walked if you are up to it, but it's a bit of a hike.
>>>Note that not all trains to Green Airport stop at Back Bay, for those trains you'll have to go to South Station, which would mean Orange Line to Red Line.<<<
HOWEVER ... on Saturday, Sunday and Labor Day, NONE of the trains go to TF Green. If traveling on a weekday, Orange Line from North Station to Back Bay will work fine, if the timing for a TF Green train is right.
Schedules are here--link is too big, you will need to copy and paste: http://www.mbta.com/schedules_and_maps/rail/lines/?route=PROVSTOU&direction=O&timing=laborday&RedisplayTime=Redisplay+Time
There are buses that run from Providence to the airport, so if you have to take the train to providence, it may still work.
Peter Pan bus runs from South Station to TF Green https://peterpanbus.com/Edited: 30 August 2017, 21:07
All MBTA (and Amtrak) trains that depart South Station and go thru Back Bay stop there. Only South Shore and Fairmount trains which use a different set of tracks out of South Station do not go thru Back Bay. To the original question, take the MBTA Orange Line from North Station to Back Bay Station to get a train to Providence or TFGreen.
Checking MBTA and PeterPan schedules from South Station. They can get you to Providence, but I see no mention of TF Green airport, PVD etc. TFGreen airport Transportation website clearly shows those lines extending ast Providence to PVD. So how to make sire the train/bus from South Station is the one that takes you to TF Green airport, not just the Providece. Thanks for any info.Edited: 26 November 2017, 04:10
It looks as though the train to TF Green does not run on weekends, but if you plug in a weekday date, you will see the departure and arrival times from South Station to TF Green.
Once you get to the Providence train station you could get a taxi or uber to the airport. Both Amtrak and MBTA commuter rail go from Boston to Providence. MBTA goes to TFGreen on a limited schedule (which is actually in Warwick), Amtrak dioes not. If you take the commuter rail it will leave you on the loading platform in the parking garage. Go to the third floor and walk across the sky bridge to the terminal.