Welcome! I'm leaving Boston for Charleston, SC so I am trying to revisit everything that I'll miss. Here's some stuff you can do under 21, mostly food related:
Chinatown: Take the orange line to Chinatown and walk around. Its like being in another country. If you like Vietnamese food, go to Xinh Xinh on Beach St. It's just across the Boston Common behind Emerson college. Go into Seamart, which is a cool asian market that has vegetables you may have never seen before, as well as a really cool live fish market/display.
Take the Red line to Harvard Square and go grab a burger at Mr. Bartley's, just a couple blocks down from the Harvard Book store. On Sundays, the Museum of Natural History at Harvard is free. Cooler than it sounds.
If you want to see some of the most beautiful women in the country, walk from one end of Newbury Street to the other. Then come back down Boylston. You may break your neck, but it's worth it.
Glad to hear you were in the North End, but Mike's pales in comparison to Modern Pastry, which is pretty much across the street. If you have a few extra bucks, go to Neptune's Oyster bar for the best Lobster roll in New England.
Go down Charles Street in Beacon Hill and stop at Cafe Vanille for awesome pastries and coffee, but don't eat it there. grab it all to go and sit in the Public Garden across from Cheers--great place to people watch and actually very pretty park with some excellent statues to check out, particularly the famous statue of George Washington at the Comm Ave entrance.
Finally, go stalk Tom Brady and Gisele at the corner of Beacon Street and Fairfield. I have seen Gisele numerous times and got a pleasant hello from him when we both went to grab a newspaper on a Sunday morning from the paper guy's truck at Comm Ave and Clarendon. Walk across storrow drive to the Charles River and take a walk on the Esplanade. It's beautiful, and there are tons of girls there, too.