Is a tent worth it? Does a duck swim?
I'm probably one of the bigger proponents of camping here. I love camping while on the road. I much prefer it to sleeping in my car or hotels.
Car Sleeping: I've done it on occasion but, like Larrison, I never sleep as well as I do in my tent. And after being in the car a good part of the day, it's really nice to get out of it. If money is really tight, you can sleep for free in your car at truck stops. Gen who posts here does this a lot more than most of us and seems to enjoy it. I just really hate it and it's always a last resort for me. The main issue I see for you is where will you sleep in the car? If there's 3 of you, there aren't even seats to stretch out on unless you have a big 3-bench SUV. Just something to consider.
At a National Park, you will pay the same fee for sleeping in your camp at a campground as you would for sleeping in your tent.
Camping: It feels great to get out of the car and stretch. I sleep like a baby in my tent. I enjoy taking a walk around the campground before going to bed. It's fun to read the license plates of other campers to see where they're from. And campers are usually friendly so I usually end up having nice conversations with other folks. Sometimes they're coming from the place where I'm headed so I pick up some nice hints. I've paid anywhere from about $10-30 for camping....most often in the $18-22 price range. And this usually includes some nice amenities like hot tubs. It takes me about 7-9 minutes to set-up camp and about the same to take down in the morning. So not a big chunk of time.
IMHO, when it comes to road-tripping, the only thing more important than the tent is the car.
Car Camping vs. Car Sleeping
I'm in agreement with Judy and Larrison about getting a tent. You'll end up much much more comforatable.
I too have slept in my car on occation, but its always an option of last resort for me. Even then, I only consider it when I'm traveling solo. I simply can't imagine trying to get a decent night sleep with more than one person in any standard size car. Even the benches of a large SUV would be pretty uncomfortable to me after a night or two.
Camping equipment doesn't have to be expensive. You can find a tent sutable for 4 people for a little as $50, and I've bought cheap but warm sleeping bags for under $10 that work fine for car camping. To me, that's a small price to pay to be a lot more comfortable.