If you want specifically Boston, I'd try Back Bay, South End or Beacon Hill. If you're open to Brookline, then I'd go with Coolidge Corner or Washington Square! Technically suburbs, but it doesn't feel like it. I live right across the street from the T, and have tons of shops/restaurants right near my house.
Beacon Hill meets your criteria, especially if you're in your mid 20s or 30s. Though lacking on bars, the neighborhood is universally loved. It's convenient for public transit to Cambridge, the South End, Back Bay, and Fenway. Just don't think about moving there with a car unless you can afford dedicated street parking or a place with a spot.
The answer really depends on your budget and how much space you are looking for. Coolidge corner in Brookline, Back Bay, South End and Beacon Hill are great (and pricey) neighborhoods. But, there are so many trade-offs to make if you want to save money or get an apartment with more space. Also, the housing in Boston is pretty old (my building is more than 100 years old). If you aren't used to that you may choose to live farther out just to get newer construction. You may want to check out walkscore dot com. It's a great resource for finding livable neighborhoods.
Ditto to all of the above and .......Charlestown is fabulous. Young and hip people, families, dogs... Brookline, Brighton, Arlington and Cambridge are all good choices too, oh, and Somerville-Davis Square is a fun spot. Good Luck!