You will want the $80 Annual Interagency Pass which lets you enter any national parks for no additional cost and gives a 50% discount on federal campgrounds:
NPS PassYou can buy it at the first park that you visit or order it ahead of time using that link.
Most federal campgrounds (nation parks(NPS), national forests, bureau of land management(BLM) and other federal agencies) run their reservation systems thru this website:
Recreation.gov
The one major exception is the Yosemite Park reservation system which is run on a lottery system five months in advance of the date you want to camp during. Unless you are in country five months early your chances probably drop significantly. Even if you are here the odds are poor. For any other major parks the sooner you make reservations the more likely you are to get them. Yellowstone for example.
Washington DC gets to be very hot during the summer, so I suggest trying to get there are early in your vacation as possible. I doubt you will be acclimated to summer temperatures in DC which can hit 30-35 deg C. The most recommended campground is Cherry Hill in Maryland. And then ride public transportation into DC from there. You do not want to drive your vehicle into DC and that advise applies to any other major city in the US. Boston and New York City for example. Once you get out of the NE many of the big cities do not have public transport so you will have no choice but to drive into the city centers. Do not pull your fifth wheel into those city centers off of the Interstate highway system. Find somewhere on the outskirts to camp.
Probably the best general route for you to follow would be I-95 south into Florida and then I-75 north to I-10 and follow that west to New Mexico where you'll probably want to get on other roads at that point.
As for National Battle Fields. Gettysburg is the best one available. There are many others all thru the south. For the Revolutionary War, Minute Man National Historical Park in Concord Massachusetts is the one to see. There are other parks of that period scattered around the northeastern states.
There are thousands of places to see in the US so we need a better idea of what you are looking for before detailed recommendations can be given. I'd get a good guide to US National Parks and read that to give you a better idea. I think you'll have some difficulty getting used to the shear size of the US. It takes most of a week to drive from Florida to California for example. And that is without a lot of sightseeing during the trip.