Absolutely no question, you need to go see a Red Sox game at Fenway. Even if you don't like baseball, Fenway park during a game is an experience everyone should have. Hell, seeing a ballgame at Fenway can make you LOVE baseball, at least live baseball. I'll tell you what, the only time I can stand to watch baseball on TV is when the Sox are in the playoffs, but I'll go to any major league game live; even a dinky team like the Brewers or somesuch. It's a truly American experience that everyone should try, if only for the energy of the crowd.
Hell, I'll even go to a Yankee game.
There's the usual touristy stuff; take a walk around The Common, and through the Public Garden, do the duck tour and the swan boats... and then there's the trolley tours which are pretty cool. Jump on and off whenever you want, and they take you around the city. Actually it's a great way to see the U.S.S. constitution, which everybody should do; and some of the various revolutionary war sites.
For the less touristy, the science museum, the childrens museum, and the computer museum (they are all within a few miles of each other) can all be visited in one good long day. Trust me on this one, the Science Museum is incredibly fun; and you can spend a whole day in there without even thinking about it. The great thing is, they're all within a few hundred feet of a T station.
Theres about 12 dozen (that would be a gross) other great museums, gallerys, and exhibitions if your fancy lies in that direction, including some of the best art collections in the world; and several of the great collections of rare books.
Fanueil hall, quincy market, copely plaza, newbury street and berkley street, and legal seafood for lunch or dinner, makes a good day if you're a shopping person.
The view from the top of the hub is frikken awesome; but the food isn't that great and it's overpriced.
If you get the chance, the boston pops is about the best pops orchestra in the country (even with Keith Lockhart instead of John Williams. Sorry, I grew up with Williams, who I happen to think is the best conductor and classical composer of the last 50 years), and the Boston Symphony Orchestra is considered THE best in the country, partly because Boston Symphony hall is considered the second best in the world after Vienna.
You can pretty much ride the red line from JFK to Alewife, get off at almost any stop, and find something good to do. Specifically JFK station has the JFK presidential library, the bayside expo center, several great parks, and three places with good live music; plus it's a short walk to castle island, which is a great free recreational park and revolutionary war fort.
In that vein, you can especially find cheap things to do that don't need a car; and cheap in Boston is always a good thing, because it's a frikken expensive city.
Also, theres only about 5000 great places to stop in for a drink, and maybe some pub grub; some with decent live music.
There's tons more I can think of, and all that's without leaving the immediate city area for the suburbs; or even the outer villages (West Roxbury etc...).
A note, Boston itself is a very small city, with only about 600,000 resident in 40sq miles, however the city of Boston annexed all of the smaller towns surrounding it over the last 200 years, which includes 4.4 million people (not including the estimated half million illegals), and about 90sq miles consisting of the neighborhoods (officially villages but no-one calls them that) of Allston, Brighton, Charlsetown, Dorchester, East Boston, Hyde Park, Jamaica Plain, Mattapan, Roslindale, Roxbury, South Boston, and West Roxbury; several of whcih contain subdivision of their own like Mission Hill, and Longmore in Roxbury; Readville in Hyde Park; and Savin Hill, Ashmont, and Lower Mills in Dorchester. Chelsea is technically it's own city (the smallest incorporated city in America at only 1.8sq miles), but most people consider it another neighborhood of Boston.Add in Milton, Quincy, Cambridge, Sommervile, Revere, Newton, Medford, Evrett, Winthrop, Brookline, Watertown, Malden, Needham, and Dedham (all the towns directly surrounding Boston and it's outer villages)... Well, I didn't run out of things to do there for 16 years. Oh and most of those places are served by decent public transportation into town.
So what about you guys? What are your Boston "Must Do" items?