Depending on your interests you have a wonderful opportunity to see some great moments in American (and thusly British) history! Virginia is loaded with historic towns--Jamestown, (one of the original settlements in the New World) just celebrated it's 400th anniversary last year, stop in at Monticello (Thomas Jefferson's home) and Mount Vernon (George Washington's home) is up that way.

Outside of Virginia, the Outer Banks of North Carolina were mention and are a good choice but may not be practical (the Outer Banks are expensive and may require you to backtrack on your route), I would suggest a stop in Raleigh instead, that keeps you on I-95 and provides you with a larger city (the capital of NC), great restaurants and good hotel choices.

Past that, there is Myrtle Beach in South Carolina. It's a tourist trap but it's one of those places it seems like everyone has been to at least once. The Canadians love it and every time I visit (I live about 3 hours away) there are tons of car from Ontario. I would recommend staying on the SC coast and visiting Charleston and then heading on to Savannah Georgia. The heart of South Carolina has no real attractions, so if you stay on I-95 you will see nothing essentially from Raleigh until you get to Columbia SC (2-3 hours).

Once you leave Savannah you will almost be in Florida. At this point my knowledge is limited (I haven't been to Florida since I was a kid) but Jacksonville FL is a HUGE city and I'm sure there are plenty of fun things to do there. Of course you will be passing Disney World, and even as an adult it's another place you need to visit at least once. Tickets are fairly expensive, as are hotels so you may need to do some research before making a stop.

After that you're almost in Miami. Remember that Florida is a very long state, its a good two hours from Jacksonville to Orlando (Daytona beach is along that route too). From Orlando to Miami is another 4-5.

With 10 days I think you could spend a day in each state and as long as you stay near the highway you have the advantage of seeing a whole heck of a lot of history and interesting sites. As someone else said get out your maps and visit the tourism websites for each state/area you will be visiting. Good Luck!