I grew up in East Falmouth, but haven't been there in a few years. Bridge traffic is to be avoided for sure. On really bad days it can be hours to get over the Bridge. I'd avoid Friday evening and Saturday morning getting on the Cape. At least you will be staying fairly near the Bourne Bridge and can avoid most traffic getting off the Cape. I'd still avoid it Sunday evening.
Although I lived in East Falmouth I've never heard of this campground. I wonder if it is new.
The Falmouth/Woods Hole bike path is spectacular. It goes along the ocean and passes a light house. A must if you like biking. http://www.capecodbikeguide.com/shiningsea.asp
There is also a small aquarium in Woods Hole. http://aquarium.nefsc.noaa.gov/
The Barnstable County Fair runs around that time and is close to the campground. Its a lot of fun http://capecodfairgrounds.com/events...e-county-fair/
Martha's Vineyard is a short ferry boat ride from Falmouth and lots of fun as well.
The movie "Jaw's" was filmed there!
The Heritage Museum has a nice collection of restored antique cars and lots of other early american stuff. Its set on beautiful grounds in Sandwich, just a short drive from the campground.
If you like Baseball the Cape Cod summer league has some of the best major league prospects playing on the cape. You can see future hall of famers play for free (at least it used to be free). http://www.capecodbaseball.org/
The Falmouth Commodores play right in downtown Falmouth.
The Flying Bridge restaurant was always a good place to eat but I haven't been in a long time. Great views over looking the $$$ boats in the harbor. The harbor is nice to visit for a stroll as well. http://www.flyingbridgerestaurant.com/
Personally I'd spend more time in Boston and skip Providence, RI, unless you have a reason to go there. There's a ton of stuff to do in Boston, no so much in Providence. In Boston you can go on the Duck boat tours, visit Fenway park even if you don't go to a game, walk historic walking trails, go to the Boston Common, view Specular fireworks if you are there on the 4th, visit Faneuil Hall Marketplace, eat at a resturaunt 50 stories overlooking the city, visit the Museum of Science, etc.
I just don't know of anything to do in Providence.