Regarding tickets, check out Seat Geek. The site compiles what is on many others sites (StubHub, TicketCity, etc.) on to one site; somewhat similar to what Travelocity does for airline travel. Plus, it come highly recommended by
@PatsWickedPissah
SeatGeek | Your Ticket to Sports, Concerts & More
As mentioned previously, Foxboro is not at all close to Boston. Do you want to experience tailgating before the game? If so, then the train is not an option as it does not leave you any time for that. The Boston Common Coach will give you a bit more time for pre-game activities.
MBTA > Riding the T > Patriots
Boston Common Coach | Schedules
If you decide to rent a car and make the drive, give yourself plenty of time; Route One is notorious for being the worst in the NFL for the amount of time it takes to get to the stadium. Also be aware that it's not cheap ($40 if I recall correctly). Park before you get to the stadium; that way you don't have to battle traffic going in the opposite direction.
Gillette Stadium - Patriots Parking
Gillette Stadium - Parking
As far as what to do while in Boston, I would highly recommend at least one meal in the North End; plenty of great choices there. Although it is a bit touristy, you should take in Faneuil Hall as well. Those two places are walking distance from one another, so you can work the two into one afternoon/evening. Or if you prefer you can do the Freedom Trail, with Faneuil Hall before or after.
northendboston.com
The 30 Best North End Restaurants (Boston) - TripAdvisor - See Restaurants in North End Boston with 136,149 Reviews
Faneuil Hall Marketplace | Shop, Dine & Experience
Faneuil Hall | City of Boston
The Freedom Trail
There is plenty else to do/visit, but at some point go to a seafood restaurant and have yourself some haddock (or if you have a bigger budget, some lobster) and a bowl of clam chowder. There are plenty of great seafood restaurants to choose from with a variety of prices to suit your budget. Newbury Street is a good place to check out if you want to spend some time shopping with plenty of cafes, etc.
Welcome to Newbury Street, Boston
Hope that all helps.... enjoy your visit!