I think Yellowstone, as suggested by one person above, would make a good spot to visit off the coasts. Depending on your schedule you can add Grand Teton NP. I would try to wait until after Labor Day (Sep 7 this year) since I think the crowds/pricing will be better but maybe others have a different idea. Probably this would mean East Coast, then Seattle. From Seattle it looks like there are flights through SLC to West Yellowstone, or alternately direct from Seattle to Bozeman (80 miles from Yellowstone).

That would involve driving around the Yellowstone area but would not stop you from making your "main" driving be from SF to LA (maybe with a detour after you pick up the car from SF, to go north to the redwoods before heading south).

There's also plenty to see between Seattle and SF; now that you have lots of suggestions you may want to try to plan out the days and see how much time you have left over.