travel during summer school (Boston)

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hi I'm in Boston now for the Harvard Summer School this is the first time for me to be in the U.S and I really want to do some travelling. Currently I want to go to New York, Washington and New Jersey. Do you have any advice about the transportation, the site to see, and most important, where to live cheaply because my travel budget is quite tight. Besides, what's your recommendation for travelling in the east? especially near Boston? And which place in Boston is interesting enough and worth going? Thanks