Was born in Medford just seconds away from Boston. Also lived in Burlington and in Lowell for many years as well. Currently live just over the border in N.H.
Apartments can be ridiculously expensive in Boston...renting just outside the city (but still very close-by) could work ($$) in places like Malden, Somerville, Medford, Cambridge. Cities and towns just south of Boston are not as desirable and can be rougher neighborhoods. If you are going to live inside the city then getting around is easy once you get acquainted. All the sports venues (except Patriots in Foxboro), schools, museums, bars, clubs, restaurants, attractions etc. are all fairly close in proximity and located everywhere of course.
edit:
You mentioned Weymouth? I have a sister in-law from Weymouth...been a few times, not a bad town. Far enough south of Boston more the rougher neighborhoods that I mentioned...but still close to Boston.