There is tons to see from the Freedom Trail, to Quincy Market, but the real gem for me is the North End. Lots of authentic Italian restaurants, Mike's Pastry's (World Famous Canolis, though the local's prefer Maria's (or something like that, just ask an Italian dude outside, seriously)). The North End is great to walk up, look at various places, and great food. Plus you can see the North Church, where Paul Revere crap with the lanterns.
+1 for the Union Oyster House.
If youre only going to be there a weekend, you should hit up North End. If you head down Commonwealth Ave, it turns into Allston, which has a lot of dive bars, and older kids. Cambridge is really cool, its where Harvard is. The Middle East is a great club over there to watch bands, and is also world famous. Cask and Flagon is cool just for the touristy stuff, near Fenway. Jillians has a 3 story bar and a lot of action down there, could be fun, and you can walk there from the North End.
The Commons is a really cool park, you go to Cheers, but its dinky and not what it looks like in the show. The barge/duck/water boat drive tour is neat. But just walk the freedom trail, its cheaper and cooler.
Chinatown is small, but pretty neat. Lots of underground gambling.
If you can make it out, Concord is a really cool little town. Walden Pond is there, and you can swim in it. HDT's cabin is there, and there is a cool little bed and breakfast/hotel called the colonial inn that is a great stay. Highly recommend. Author's ridge, and sleepy hollow are there. Salem really isnt worth the visit. Its pretty dumpy compared to all the other towns.