Welcome to RTA!
To be perfectly honest, we could all throw ideas at you, but perhaps they wouldn't interest either of you. My advice would be to dig through this website then open up a map to see what type of routing you could take. Once you get some better ideas of the things that interest the two of you, you can come back here and we can help you fine tune.
Personally, staying in the car overnight is neither safe nor a good idea unless it's an absolute emergency. You just won't get a good night's sleep in your car in a parking lot someplace! It would be better to borrow a tent and a couple of sleeping bags, grab a cooler and some food, and do a little primitive camping. Your body would much prefer sleeping on the ground to sleeping in a car.
AAA/CAA is *always* a good idea. The price of the membership will be paid for two or three times over if you need a tow or a battery jump-start in the year you have the membership. Besides, you can also get maps and materials free, that will help you plan your trip.
BTW, a planned trip or a "go as you please" trip can be just as much fun. It all depends on your traveler profile, which as beginners, you are just developing! Ten days can get you about 1500 miles before you'll need to come home, trying not to have marathon driving days.
Oh - - you're not crazy. Because if YOU are, all of us who love the road are crazy!
Donna