Just back from 3 weeks driving round California. Definitely do the trip! We had a Challenger & I’d agree that a modern car is the way to go. Just be aware that luggage capacity is limited for the size of the car you’ll get. 2 of us in the car & the boot/trunk was just big enough for 2 suitcases & hand luggage. Great car though & got comments from locals.
For a cross country trip I’d say 3 weeks is the minimum, there’s so much to see & do you’ll regret having to drive on to meet your targets.
I’d second the suggestion about driving down the Appalachians. Did that a couple of years ago and loved the drive. This trip we passed the end of 66 and I would recommend Barstow as a stop. Big train yard & a couple of museums to visit. I’d go back there again. Just make sure to be there at the weekend as they only open then.
I love driving in the US, especially when you find a small town to stop in. We traveled booking hotels on Priceline or Booking.com mostly just a day or 2 in advance which have us the flexibility to change our plans a little. An RV would be great if you plan to stop in parks, but as said, the big parks need to be booked well in advance.
Already started thinking of our next trip…