From Logan Airport take Silver Line L1 to South Station and from there the Red Line into Harvard, or anywhere else you want to go. Silver Line is free of charge, and so is the connection to the Red Line.
Taxis from Logan Airport into Cambridge are around $40.
Local transportation: pick up a Charlie Card for free in one of the major subway stations (Harvard, South Station, Alewife). Fill and re-fill in automatic machines. Charlie Cards are used for all subway and bus lines. There are no student rates. A monthly ticket granting free access to all transportation is $85.
Uber and Lyft both operate in the Boston area, and are sometimes the fastest and cheapest way to get around.