To be honest I wasn't that impressed and you will be giving up a lot of time to make it that far east. The Grand Canyon is impressive but again you give up lot of time getting there. Can you afford to fly? Driving in the desert is pretty at first then is just goes on and on. Keep in mind you will be there in August and the call it Death Valley for a reason.
There is so many amazing places in California you could spend a month there and still not see them all. Hell I could spend 9 days in Yosemite. I have driven pieces of HWY 1 but never the whole thing so I can't tell you how long it will take. It will take longer than the interstate but what I have driven is was awesome. It really depends on how you want to spend your time. In my opinion if you want to see America you need to stay off the interstates as much as possible.
But hey if you want to really get the American experience make sure you spend some time in Compton and try and do it late at night walking.