Mountains or seashore? Cities or small towns? Renting an RV may be hard in Washington DC. Would you consider taking a train to a place where you can rent and make a loop and then return to Washington?
For the seashore I like the Outer Banks of North Carolina.
For mountains western North Carolina and eastern Tennessee have the Great Smokey Mountains.

A couple of things to consider about travelling in a camper/RV:

1. Being from Europe, are you comfortable driving the roads in the U.S.? I would suggest that if you haven't driven in the States before, you shouldn't start out by driving a camper. It's a very different kind of driving.

2. Are you comfortable with the idea of driving and living in the same vehicle for 14 days, especially with two young children? Personally, that would drive me nuts. I would want the variety of a hotel or cabin every few days.

Thank you all for the tips!

Yes, we are quite comfortable driving in the (I mean, how hard can it be, same drivingside as we are used to and people drive more relax in US than in our country, it's not like in north afrika or Italy with the typical chaotic driving style - we went a few times when we had no kids yet, of course, we never drove a camper though...

Good tip about combo RV and hotels, also, we are absolutely willing to travel by train a bit and rent an RV for a week and maybe change cars...

Also, we prefer small towns and typical not too touristic/crowded places

DC with kids is a really good tip, we ill buy that book!

You can rent a 25' class C motorhome from cruiseanerica in Richmond for about $700 for a week. http://www.cruiseamerica.com/
You can find hotel/motel rooms for less than $100/night and not have the hassle of paying for more fuel.
The Amtrak fare for your family is about $70 from Washington to Richmond Virginia.

http://www.buschgardens.com/bgw/
http://www.history.org/

Many of the campsites have small rental cabins. Campsites in the U.S. can be really great for the kids - waterslides, activities, ... Check out the Jellystone campgrounds. A little pricey but you get a lot for the money. My kids love them.
We love a place in PA called Raystown. Beautiful lake and water park.
Do you plan to go north or south from D.C.?
Harpers Ferry and Gettsyburg are two nice day stops if your heading north. PA has lots of beautiful places to camp.

Not sure where you are from - but the US may well be significantly hotter than you are used to - esp if you head south from DC in July (temps in the 90s are frequent and 100 not unknown) - so take that into consideration in your planning.

Also - RV camps and camping spaces are limited - and many reserve far in advance - so be sure to take that into account. Finally, watch for roads usually called "parkways" - these typically are for cars only - and can't be used by any larger vehicles (not trucks, buses or campers allowed).

>>Finally, watch for roads usually called "parkways" - these typically are for cars only - and can't be used by any larger vehicles (not trucks, buses or campers allowed).<<

Please disregard this misinformed advice. There are some scenic roads called parkways which restrict commercial traffic but the majority do not. Rest assured there will be signage for any road that restricts any type or size of vehicles before you enter the road.

Welcome, Katleenkids. There is so much to see in D.C., you could almost spend the entire time there. Smithsonian museums have so much to see. Indian and Air and Space are very popular.

If you wish to go to the ocean (Atlantic) beaches avoid weekends as traffic can be miserable.

Annapolis, Maryland is an easy drive from D.C. Nice town for shopping and eating. Naval Academy has a very interesting tour and museum. I don't have children but there's a pirate ship that some kids like (tp://www.chesapeakepirates.com/). Of course you have to eat crabs if you come to Maryland!

You could conveniently drive from Annapolis to beaches in Delaware or Maryland. Again, watch for weekend traffic.

Williamsburg, Yorktown VA are interesting also if you are interested in early American life. Yorktown has a beach.

Baltimore Md. has a nice Inner Harbor area. An Acquarium and a Science Center the 6-year old would enjoy.

The campers I've seen in Italy looked much smaller than the typical ones in the U.S.-I'd be leery about renting one. Driving a very large vehicle in an unfamiliar area doesn't sound like fun to me.

Good luck deciding.