Head down to Fenway Park where the Boston Red Sox play. Coming from New York to Boston University (which is loaded with New Yorkers), it was/is fun to root against the BoSox and for the Yankees (although the Mets always get my cheer against the Yankees).
Fenway Park is home to the Green Monster in left field - a thirty foot wall making it super-hard to hit homers. It is also one of the oldest remaining stadiums in the US with a manual scoreboard at the bottom of the Green Monster.
The cheapest seats are $20 for bleachers, making the park also one of the most expensive to go to. BIG TIP: DON'T BRING A BACKPACK. They don't allow them in the park, and you'll have to check it at the Gold's Gym across the street for $10. Too much.
I go more for the hot dogs than the homers though. Also, be mindful of the following: Babe Ruth played for the Red Sox until 1920, when he was sold to the Yankees. The Sox haven't won a world series since, while the Yankees have won around 1/4 of all the world series since.
Go visit the historical site of Fenway Park and once you are there ask the guide or a Boston fan who Buckey Dent and Aaron Boone are. I am sure they will be more than happy to answer your question
Even if you don't have tickets to the games, go to Fenway Park on a game day, just to enjoy the food and watch all the people!