Boston is a city that I absolutely love traveling to! I travel to Boston about once every 6 months and have an incredible time each Im there.
My first stay I took in the tourist sites. My top recommendation to get a feel for the city would be taking the Old Town Trolley Tour. http://www.trolleytours.com/boston/
Tickets are about $36.00 and it is a hop on hop off tour so if you see somewhere you would like to get off its that simple!
One of my favourite things to do in Boston is to catch a Red Sox, Celtics, or Bruins game.
I also recommend setting aside an entire morning to just walk around! Boston is a city that is fairly compact and easy to get around. Beacon Hill has some beautiful homes and the North End as well. There are little bits of history throughout the city including Paul Reveres home, cemetarys and the Freedom trail the goes throughout the city.
The bars and pubs in the area are a lot of fun, but a warning you will need your passport as ID if you are from out of state...it may be out of country. But I definitly needed it being from Canada, my drivers license wouldnt work. And most places ID everyone, even if your mid to late 30s.
There is also lots of great shopping. DSW at downtown crossing is incredible, floors of designer shoes at amazing prices! Newbury Street is also a popular spot if looking for designer shops.
My favourite restaurants include Eastern Standard (Back Bay). The Ivy (Downtown), Jer-Ne in the Ritz-Carlton (Boston Common) and for a more casual meal Intermission (Theater district). Southstreet Diner is also great as its open 24/7 for those late nights. Durgin Park (Fanueil Hall) is a Boston trademark and I swear they have the BEST Boston Cream pie!
My favourite pubs and bars include Whiskey Park and the Estate on the more upscale side and the Greatest Bar for those post sports game nights. There are a lot of other great options as well.