Hi, CJ ~
It depends on your budget and what's important. Coolidge Corner in Brookline, the North and South Ends of Boston and Back Bay are fantastic as you are near just about everything from grocery & department stores, restaurants, clubs, night life and public transportation while not being overrun by undergrads. The C branch of the Green line T is a big attraction, a little more "adult" while allowing you to get everywhere in the city with relative ease - you can live just over the line in Brighton, providing the same access to everything mentioned above while also being a little cheaper.
We at City Central Realty, LLC in Brookline specialize in rentals and sales throughout the Greater Boston area to recent grads and young professionals.
Please let me know if there's anything we can do to help you in your search.