Since I'm a native too, I'll kick in my three cents worth.
First, the Chesapeake Bay Bridge-Tunnel is a must-do, just for the sake of it. It's not especially scenic or breath-taking, but it's the only one like in North America, and is worth the $10. Going North, it will also dump you on the Eastern Shore/DelMarVa peninsula which is a quiet, interesting drive up to the Delaware Valley. (I-64 around Norfolk can be a b**ch, but you'll live.)
If you're into that stuff, Williamsburg and the other "colonial" sites and sights in the Mid-Atlantic can be interesting. Monticello, Mount Vernon, etc. I won't cover DC as there are other much more experienced people here who can tell you about it.
The big problem over here on the east coast is that there are not a lot of RV parks, or campgrounds in general, except in real tourist-attraction rural areas. And certainly none around any of the 'downtown' areas of the older East Coast cities, that I know of. People use Liberty Park for NYC, and Cherry Hill for DC, but I'll be damned if I can recommend an RV park anywhere near Philly.. :) Plenty of T/As and Pilot/Flying Js and things across the river along I-295 and the NJTP, but I don't know of anything on the western side of Philly. Too built up. I'm not even sure if there's a good RV place out around Valley Forge. There's a KOA out near West Chester, 35 miles west of Philly.
There's fairly decent mass transit in the city proper of Philly, but your problem will be getting to a station. If you find a place over on the Jersey side near Camden (ugh), you can get on what is known as PATCO, which is a high speed, few-stop commuter train that will get you right in to Center City Philly in about 15 minutes from it's farthest station. Camden has a big aquarium along the river, and the USS New Jersey battleship, but not much else to recommend it, unless you like viewing decrepit old refineries and abandoned row houses. You can get around much of Philly on foot, or a quick taxi ride is easier than trying to figure out where the SEPTA buses might be going this time. Independence Hall, the Bell, City Hall, the Franklin Institute, and the Art Museum are typical tourist destinations, but check out the Mutter Museum, Penns Landing, and some of the other places near center city. The old Eastern State Penitentiary is just a block or two from the Art Museum, and is an interesting tour for the penal system aficianado.
It might almost be worth renting a car for the day or two that you visit Philly. The traffic here is nowhere near as hectic or frantic as around DC or NYC, just don't get caught on the Schuylkill Expressway at 5:15 pm on a Friday night in the rain... :)
Another option is to park at one of the newer casinos near center city Philly - I think SugarHouse allows overnight large-vehicle parking. It's on the waterfront. Harrahs/Philly is actually in Chester, about 15 miles south on 95 along the river, and you don't want park overnight there. Trust me. There's also Parx out near Bensalem, in the northeast of town, that I think is also fairly close to mass transit.
Here's a quite useful blog that has more info.
http://www.cheddaryeti.com/2012/06/20/rv-parking-for-philadelphia-10-reasons-its-worth-your-effort/