For inexpensive overnights, there is Couchsurfing.com, not in my comfort level, Campgrounds, but pricing and restrictions vary. Some are listed as seasonal only, some will not accept tents, some have minimum number of night reservations, especially around holiday weekends. It is not unusual for State park campgrounds to have a reservation fee from $5 -$15 per reservation. Airbnb is a great option in many locations where options are few. you can usually find hostels in the cities, Washington DC has one reasonably close to the main attractions. Finally hotels and motels.
You might want to include the state in several of the cities listed, as several towns with the same name in the area but in different directions and may not be the ones commonly thought of. Cumberland is usually thought of as the one in Maryland at the C&O canal- Great Allegheny Passage rail trail trail, north west of Washington, not the Virginia. Newark New Jersey is more well known than the Newark Delaware which I believe is on your route.
Going by your towns listed, from Washington to Charlottesville you are heading well into the mountains, you might want to look at the Skyline Drive and Blue Ridge Parkway, These are lower speed limit scenic roadways but can be very hilly! and are close to your listed towns.
It looks like you will be going up the Atlantic coast from Norfolk. You will need to catch the shuttle across the Chesapeake Bay Bridge Tunnel as cycling is not allowed for safety.
Crossing the Susquehanna river at Harve De Grace, North East of Baltimore, is unknown as several years ago there was a pilot program to allow cyclist to use this bridge during limited days and times, the nearest crossing was west near Berkley on US 1.
As for must see, I definitely recommend the Holocaust memorial/museum in DC. The Air & Space Museum both in Washington, DC and the one next to Dulles airport, the Steven F Hdvar-Hazy Center. Any of the other museums/sites depending on your interests while in Washington, DC. It is very easy to spend days visiting all the sites in DC alone.
Near you route, near Norfolk is Colonial Williamsburg, a 1600-1700 recreation site.
When do you plan on touring? Finding accomidations late May (Memorial day weekend), Early July (July 4 Independance Day) and Early Sepetember (Labor Day) can be tricky and costly. If you are touring on July 4th you will find most towns and cities having firework displays.
