It sounds like you are doing a historical tour of Eastern USA. The most convenient way to get to all the sites is via rental car. I would definitely advise you to at least rent a car in each of the major areas you will be visiting. If you prefer not to drive the longer distances, you can commute between cities via train or airplane. Fortunately, all the areas you will be visiting are along the Amtrak East Coast train route. They are also serviced by commuter air routes. You should be able to find reasonably inexpensive air fares as well as reasonably priced train fares.
If you begin in Boston, rent a car for a few days. It will make it easy to visit Plymouth and the Mayflower.
I don't know where Avondale PA is located, but you can drive to Philadelphia, take a train or an airplane. I suggest you rent a car in Philadelphia for a few days. From the city, it is a short 30 minute drive to Valley Forge, and only an hour to Gettysburg National Battlefield.
You can drive from Philadelphia to Richmond or you can again take a train or an airplane. You should definitely rent a car in Richmond for a few days. Jamestown and Williamsburg are less than an hour drive east of Richmond. Charlottesville and Montecello are about one hour west of Richmond.
If you rent vehicles in each of these three locales, you can easily find reasonably priced accommodations on the outskirts of each city. There are plenty of hotel choices. I like the Hampton Inns. They are medium priced, always clean and safe, they provide very comfortable beds and a great free breakfast bar.