I'm not an expert on BOS but I believe that the T (aka metro) goes out to the airport. The T is really easy to understand and easy to ride.

In NYC, getting to JFK is a bit more of a hassle. The easiest is to take a taxi - however, this can be expensive, and take over an hour at rush hour. The subway (A train) is cheap but can be crowded, it's a schlep with luggage and the A train goes thru not so nice neighbourhoods, then you have to switch to a bus for the last bit to the airport. From Manhattan, this will always take you well over an hour.
You can also take the AirTrain to Jamaica station and then the LIRR from there into Manhattan (Penn Station) and vice versa. From Penn Station you can easily catch a cab or the subway. Always depends on how much luggage you have.