Firstly, congratulations to your wife for being accepted to a Simmons graduate program.
The MFA is indeed free to students, but that would leave you to pay for a ticket. They also have free admission Wednesdays after 4 pm, though they request donations.
I would wager that the public transit in Massachusetts is superior to that found in most other places in the nation. It's easy to make your to the city from the suburbs and vice versa. It can add up, but there are monthly passes, and it's quick and convenient. Can find more info about the MBTA here:
Located in Jamaica Plain is the Arnold Arboretum, which is quite beautiful and also free. For an idea of what you might see, you can check out this flickr album they advertise on their site. It's all content from visitors.
https://www.flickr.com/groups/arnoldarboretum
http://arboretum.harvard.edu/visit/hours-services-and-policies/
Hiking: http://www.mass.gov/eea/agencies/dcr/massparks/region-south/blue-hills-reservation.html
You may want to check out Fanueil Hall: http://www.faneuilhallmarketplace.com/
You're coming to town at an ideal time, as there are Italian festivals (known as Feasts) that take place during the summer in the North End. The most prominent is arguably St. Anthony's, which will occur on Labor Day weekend. Streets are blocked off, vendors line the road, there's live music, and people are everywhere. While there, you might want to grab some pizza from Pizzeria Regina, and grab some desserts from either Maria's Pastry or Modern Pastry. More info to see what you can expect here:
http://www.stanthonysfeast.com/
Regarding beaches: you'll probably want to visit many of them, but you could start with the tourist-y thing by knocking out a visit to Plymouth Rock while heading to the beach there. Wood's Seafood is a hole-in-the-wall place located right on the water that a lot of people love.
It goes without saying that you might want to catch some games at Fenway or the TD Garden, provided you're willing to pay out the ass for tickets and food. There are also many, many places in and around Boston to catch concerts. Also keep in mind the many universities/colleges around. You'll often find theatrical productions put on by students that are enjoyable and far cheaper to attend than a professional performance.
It's understandable being freaked out about moving to a new place not knowing anyone, but I think you'll enjoy your time here. It's a great city, and there's literally something for everyone. Massachusetts is very much a place rooted in our nation's history. As I said before, visit Plymouth Rock. Visit the Old North Church. Embrace the city, and have fun. :)