There are a few breweries in the city that you can visit. Trillium does farmhouse beers, Idle Hands does Belgians, Night Shift does mostly sour and Brett stuff, Mystic does saisons, Harpoon is more run of the mill but they have a fun beer hall. Aeronaut is a new one that just opened up and is run by a few MIT grad students. If you can get out of the city, Jacks Abbey is in Framingham.
For bars, I'm more familiar with the Cambridge side since that's where I live. I like Bronwyn a lot for hard to find German beers and great food. Lord Hobo if you want a draft list of 40 beers that are mostly IPAs. Meadhall has around 100 drafts and there's typically at least a few good things on tap there.
There's lots of great food here. Some of the best cheaper hidden gems that I really love are the Druid in Inman Square for the best fish and chips I've ever had; Cafe Mami for Japanese pork katsu curry on par with anything I had in Japan; Coast Cafe in Cambridgeport for truly mind blowing southern style fried chicken. Clover is a unique vegetarian/vegan place of you're looking for something healthier and they have several locations around the city.
In North End, skip mikes pastry and modern and go to Marias. She closes early, though. I like Neptune Oyster in North End a lot as well, but you pretty much have to go there at lunchtime to put your name in for dinner. Regina Pizza and Pomodoro are other good Italian spots in North End.
Don't miss Flour Bakery. You should get breakfast, lunch, or a pastry and coffe there at least once during your visit. I think if I had to recommend one single can't-miss place, Flour would be it.
If you have some money to burn, go to Oleana or Craigie on Main. Salts is another favorite of mine but I think they're still closed for renovations. I often feel that I don't get my money's worth at a lot of high-end restaurants, but I think that these three really are worth the cash.
Those are a few of my favorite places just off the top of my head, but they're just the tip of the iceberg.
Oh, and if you want to see some real killer pickin', check out Duke Levine and Kevin Barry, two local guitar heroes. They play at the Lizard Lounge pretty regularly.
Welcome!