Fanuel Hall is best for the tourist
Newbury Street in Back Bay or Charles Street in Beacon Hill
Wander down Newbury Street for the best and most-varied array of stores in Boston.
Newbury Street or a stroll around the South End, Motley in particular. Union Square in Somerville also getting really good, but a bit of a trek
Newbury Street is great for shops, and Copley Mall has all the fancy high-end stores ticked off - Barneys is up there.
Harvard Square, in Cambridge
Newbury Street is great for shopping, or the area near the Boston Common