Everything in the Boston area is within 30 mins of Charlestown. I am not sure why you were told to stay away from the Green Line. it is fine.
You can try the North End, East Cambridge, Back Bay, Beacon Hill, West End, and Midtown.
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That would be wise considering the Green Line doesn't go anywhere near Charlestown (with the exception of North Station). Try the Orange Line instead. Forest Hills, Stony Brook, Green Street, to Bunker Hill C.C. in Charlestown would take about 30 Mins.