Flight
TF Green (PVD) is your best bet, 40 min drive to Bristol. Logan in Boston (BOS) is also doable, 80-90 min drive to Bristol.
Travel Note
Bristol, Warren, and Barrington are closest to the wedding and have AirBnB, VRBO, and Bed & Breakfast options. Providence has many options for rentals and hotels. Close proximity to TF Green Airport (PVD). Fantastic dining and many things to do. South of Bristol you have Portsmouth, Middletown, and Newport which have both rentals and hotels and are great for a long weekend.
Hotel
~5 min away Limited availability, but some rooms available so book sooner rather than later