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Seriously you WON'T want to drive in NYC. The subway is good enough.
Haha come to Washington DC!! I am studying in Baltimore, MD which is an hour away :) Though if you were to come to DC only, honestly I don't think you'd need to join any tour... Most attractions in DC do not charge admission fees, and there are dirt cheap ($20 one way) Chinatown buses from NYC Chinatown (88 East Broadway) and NYC Penn Station that goes to DC Chinatown. Some of those buses go to Philadelphia also!
One of those service providers:
www.apexbus.com
I am not familiar with Philadelphia, so I wouldn't comment on that...
If your main focus is New York and the East Coast cities, it would be hard to fit Niagara Falls into your itinerary (it is out of the way - need to fly 1.5 hours to get there). If you have only 10 days, I would suggest 2 days in Boston, 4 - 5 in NYC, 1 - 2 in Philadelphia and 2 in Washington. Don't forget, it takes 4 hours to get from Boston to NYC, 2 hours from NYC to Philadelphia and 2.5 hours from Philadelphia to Washington so, plan that in as well!
Anyway, this is my own itinerary when I first visited NYC; I went over Christmas and part of my purpose is to meet friends so you should take that into account:
Day 0: Arrival
Day 1: Columbia University (am), The Metropolitan Museum (The Met) (pm), Time Square (evening)
Day 2: Lincoln Center/Columbus Circle (am), Chinatown/Brooklyn Bridge (pm)
Day 3: United Nations (am), Grand Central Station (pm), Wall Street/Ground Zero(pm evening)
Day 4: Battery Park, Statue of Liberty, Ellis Island; Rockefeller (Christmas Tree) (evening)
Day 5: Empire State Building (am), Natural History Museum (pm), The Nutcracker (Ballet) (Evening)
Day 6: Central Park (am), New Year Countdown (pm - evening)
Day 7: Return to Baltimore
I didn't go to some popular places like Madam Tussaud's, etc mainly because of cost. Though if you buy the New York Pass thingy it should give you pretty good deals to everything.
have a nice trip!