Velkommen! Welcome aboard the RoadTrip America Forums!
First, let's direct you to this post describing a coastal route, among others, up the east coast. For far more detail and many, many more ideas you can check out the old Ocean Hiway. In Florida, I'd probably stick to I-95 as the coastal routes, US-1 and FL-A1A, are heavily built up and slow going - but the Kennedy Space Centermight be worth a visit. I'd also be sure to include stops in Savannah, GA and Charleston, SC. I think your plan to drop the rental car off when you arrive in Washington and take the train to New York is a good one. A car can actually be a hindrance in both cities, which have good subway systems and limited parking, and train service between the two is quite frequent and relatively reasonably priced. And yes, a week is a fair amount of time to drive the coast providing you don't want to spend multiple days in a lot of cities.
AZBuck