Your overall trip can be done, a bit rushed but not to bad. Not much camping until you hit the Badlands.
The east is more expensive, New York will be about $100 per night for a hostel, maybe $150 for 2.
Decent motels will run about $250 a night, you still have to eat and pay for activities.
Chicago won't be cheap either.
Rent 1 car Philly to Pittsburgh, fly to Chicago and when you leave rent another car and drop off in LA. The drop off fee could be pricey.
You'll need 1 or 2 cooler/ice chests so bring them on the flight over and stuff them with camp gear. With 3-4 people you might look into a car top carrier to open up space in the vehicle.
Camp gear, you can buy basic throw away gear at most large box stores, about $200 or so for 2 people.,
2 sleeping bags, 2 pads a tent, stove and some other basic cheap crap.
Personally I would recommend dropping the Pittsburgh to Deadwood part, kind of boring but I guess some ones into purple. Saves 10 nights.
Add DC for 2 nights, NOLA or Chicago for 3 nights ( music and night life). Then fly to Las Vegas and rent a vehicle, it's soooo much cheaper for the drop off fee, maybe nothing. Vegas pick up and LA drop off is very competitive.
Las Vegas to Yellowstone is about a 2 night drive, nice and scenic after SLC and then the trip back. NO double backing and you add about 3 nights for playing.
Number by the place = nights
New York - 4
Philadelphia - 2
Pittsburgh - 2
Chicago - 3
Madison - 1
Minneapolis - 1
Mitchell/Corn Palace -1
Badlands NP-Mt Rushmore - 1
Deadwood - 1
Cody - 1
Yellowstone NP - 5
Grand Teton NP - 1
Salt Lake City - Dump SLC and go to Dinosaur NM - 1
Moab - 2
Boulder - 1
Zion NP - via a half day in Bryce - 2
Grand Canyon & Skywalk - Go to the north rim and camp on the edge in the NF, Dump the Skywalk, very expensive, out of the way and NOT the NP. - 2
Las Vegas - 1
Los Angeles. - 3
Oh yeah, at your first National Park get an annual Pass $80 bucks and good for entrance for a year to all NP's and monuments, saves a few bucks. Still have to pay to camp.