There is nowhere on earth like Fenway Park on a summer evening, and if you haven't been, fix that before you do anything else on this list. After that, there's plenty more to keep you busy in a city that takes sports the way other cities take religion.
The Cathedral
Fenway Park opened in 1912. The seats are narrow, the rows are close together, the Green Monster looms over left field like a dare, and a Fenway Frank from a cart outside the gate before you walk in is the correct move. Don't overthink it - just get one. Sit in the right field bleachers (Section 42 gives you the Monster view, Section 39 catches afternoon sun), get a Sam Adams from the vendor, and watch baseball the way it was meant to be watched.
Above North Station
TD Garden is one of the more practical arenas in any league - you can take the commuter rail directly, which means skipping the parking disaster entirely. The Celtics and Bruins share the building, and both franchises have been genuinely good recently, which means the crowd actually shows up ready to make noise. Bruins playoff games especially: the Garden gets loud in a way the building doesn't seem large enough to contain. Get the nachos from the North End cart stand on the main concourse.
The Pilgrimages
Gillette Stadium is a 90-minute drive from the city (budget more on game day), but Patriot Place - the outdoor entertainment complex attached - gives you a real pregame alternative if you don't want parking lot chaos. Get there two hours early.
The Boston Marathon in April is its own category. Post up near Mile 25 on Comm Ave (the crowd is loudest there), or get to the Boylston Street finish line area early for a spot on the barriers. It costs nothing and it's unforgettable. For ticketed events, Evil Tickets has seats across all the major Boston venues.