Great plan! I did that in two round trips, one from San Francisco to Texas via Redwoods, Lake Tahoe, Nevada, Mono Lake, Death Valley, Arizona (Grand Canyou, Monument Valley, Phoenix, Tucson, Ajo, Bisbee, Tombstone, then New Mexico with Santa Fe Madrid(!), Alamogordo, Roswell to Carlsbad Caverns and back. The second lap was NYC to Texas via Blue Ridge Parkway (Virginia) North and South Carolina, Georgia, Alabama, Mississipi along the Gulf Coast and Louisiana, then back through Tennessee etc.
I did the first lap in December/January which was fine with the weather, still warm. The second lap was in November and was a bit chilly in the north.
There are really many things to see and do.
Most impressive to me were San Francisco, Yosemite and the Mono Lake. There is a mountain pass connecting both, but only open in the summer.
In San Francisco, there is a 70 miles scenic drive which takes you to all the famous places, an absolute highlight in an open convertible.
I liked the desert along the Mexican border from California all the way to Texas. There are some very remote dirt roads.
New Mexico is great, Santa Fe and the mountains to the north are nice, the White Sands are quite spectacular when itīs a beautiful day.
The Gulf Coast is nice around Mobile and to the east.
The Blue Ridge Parkway looks very European, but is nice too.
One of my absolute favourites is Long Island east of Manhattan. September and especially October is great there. In the eastern half, the southern beaches are fanatastic (Fire Island) and further on to Montauk point. In the north, there is a great Hotel and Restaurant (
http://www.danfords.com) in Port Jefferson at the ferry port to Connecticut. Try the lobster, boiled, not grilled. ...
Driving across the US is not problematic except that you should have sufficient supplies of water, food and blankets or sleeping bags while travelling in remote areas. Fuel supply sometimes is an issue, so if you enter a desert or the death valley, for example, fill up.
And if the cops stop you, remain in the car and keep your hands visibly on the wheel!
Technically, driving an old car is no problem. For the trip from NY to Texas and back I had a 200$ Plymouth Fury station wagon of 1970 and only needed a battery and an alternator, everything else worked ok.
I have to stop now, or Iīll have to book a flight!