We came back 2 weeks ago from similar holiday, 3 weeks, with 17 & 19yo. We booked flights, car hire and some hotels with Trailfinders, also booked some hotels and a cabin at Yosemite direct. Will you do car/hotels or a camper?
I got quotes from a couple of other agencies, then we sat and went through a detailed itinerary with Trailfinders, putting together our ideas and those from other companies.
We flew into San Fran and out of Vegas, which means it isn't too hot to start with - bearing in mind that Death valley gets up to 50 deg (was about 48 when we there - v oppressive) and Vegas only slightly less. The main accommodation in DV is at Furnace creek - as DH said, it is aptly named!
We did 4 nights SF, then Monterey and Cambria on coast (supposed to drive Big Sur between those places but it is currently closed so we did part of it from Monterey, rest from Cambria). Drove inland to Yosemite for 5 nights, then Bishop, Death valley, Henderson (for Hoover Dam), grand canyon, back to vegas. None of us was bothered about LA, but a friend who did similar last year did SF, Yosemite, Monterey, down coast to LA, across to Vegas and grand canyon.
We found it tiring, both doing the driving, but also the heat (did a similar road trip in Canada in 2013 but was cooler so easier).
Whilst 5 nights in Yosemite sounds excessive, we stayed in a self catering lodge, which was lovely after hotels, and just chilled on first day. And there is plenty to do in Yosemite.
I would say plan your route as a starting point, then talk to different agents about hotels etc, even if you book yourself, you get a variety of ideas. Look at mileages, don't try and do too much in any one day and maybe have a chill day somewhere.
Some things to bear in mind - accommodation actually in Yosemite valley gets booked up a year in advance, so book early if that's your choice. Our lodge was 20 miles out.
We picked up hire car at SF airport after our 4 nights, would not have liked to have driven in the city.
Alcatraz booking opens 90 days in advance and must be booked. Worth going, we did an evening tour which was good, but given we were jet lagged and waking early, would have been better doing a morning trip.
Feel free to ask me more questions or PM me. I also went in the 80s with my parents, on a coach tour, Vegas is now a completely different place, and I don't remember SF being quite as hilly!!!