You would not need -- or want -- a car in Boston or Washington. Both cities have extensive public transportation systems, and parts of each are certainly walkable.

Washington is likely to be hotter -- perhaps uncomfortably so -- than Boston in July.

Depending on one's interests, there is plenty to do and see in NYC for eight days. But if you want to do a two-city trip, I encourage you to look into using Amtrak between NYC and either BOS or WAS. The train takes you from city center to city center. On a random weekday in July, Amtrak is offering a (non-refundable) advance-purchase one-way fare from NYP (New York Penn Station/Moynihan Train Hall) to BOS (Boston South Station) or BBY (Boston Back Bay Station) of just $28 USD per person. The trip takes a little over four hours. (The trip to WAS is a little shorter; the discount fare is $29 USD.)

And if you wanted to make an en-route stop -- say, in Philadelphia on the way to WAS, or in the seaport city of New London, Connecticut, on the way to BOS -- Amtrak permits free stopovers of less than 24 hours on its tickets. Use the "multi-city" option on the Amtrak booking engine if this is of interest to you. Plus, if you do make a stopover and do not want to deal with all of your luggage for one night, you can check some of your bags from NYP through to either BOS or WAS at no extra charge. They should be available when you arrive at your final Amtrak destination the following day. (The bags would be transported on the overnight trains, or one of Amtrak's other trains with baggage cars.)