If you are there in September, make sure you get out of the city! It's much too beautiful that time of year to stay in the concrete jungle.
I grew up 30 miles to the west of the city, in the shadow of the venerable Mt. Wachusett and this area has that simple New England charm - town fairs, old-old-old graveyards, houses and families that have been there since the beginning of your American History textbook... it's something we don't get to fully grasp here in the west. I think it's remarkable.
But while in the city:
A. Do not attempt to drive, take the T.
B. Eat... anything in the restaurants of the North End, as heas been mentioned, but also get pizza and grinders wherever you are until you can't take anymore because there's nothing comparable out here.
C. Boston Common, the Public Gardens, the Swan Boats and all of that area is pretty good and gives you easy access to the Freedom Trail (the Bull & Finch is a waste of time though).
D. Quincy Market and Fannual Hall, along with Haymarket Square nearby are all over-rated in my opinion. Pike Street in Seattle is every bit as good...
E. Drink in the accents... one of the priceless sources of unintentional commedy that you can't touch out here, though I find when people say "Worshington" and "Maysure" (Washington and measure respectively) pretty funny as a part of life out here...
F. If it interests you, visit the colleges... Harvard and MIT off the Red Line, BU and BC down the Green Line... plus a fistful of others scattered about the area
G. If you can get in on a tour of Fenway, don't pass it up... if you can't get tickets, haggle hard with the scalpers... like to the point where you think they might take a swing at you... you'll get your tickets.
H. Harbor Cruises to places like George's Island are a good way to spend an afternoon
I. Don't know the exact demographic you fall in, but there's live music all over the city every night of the week, especially if you spend time in the Allston/Brighton burroughs... lots of kids getting drunk, which can be fun if you're into it. There's also a bar down Commonwealth Ave (the name escapes me right now) that has like 50-zillion different beers... taste each one, then take a cab back to the hotel.
J. For shopping, Newbury Street and Copley will keep the wife very busy... again, skip Fannual/Quincy if you can.
K. Street Musicians in and around Park St. Station and Government Center are one of the things that adds genuine charm to walking the city.
L. Depending on how late in the month you are there, the later the better, think about figuring a way to get to Cape Cod - I think you can take a ferry from Boston to Provincetown.
M. Check out the State House, MFA, Aquarium and Science Center as time and funding allows... I would almost say, in that order.
Alright, that's quite enough.
I am impressed at how much has already been shared on the thread here... are there really that many of us Northeastern Transplants on iFish? Wow.