Wherever you fly into, have a couple of days to get over the flight before you pick up a hire car.
Look at the car hire brokers like RentalCars and HolidayAutos and book through a cash back site where you can. An SUV will let everyone see more stuff when travelling.
Don’t underestimate distances - we did about 3500 miles in just under 4 weeks (San Diego - LA - SF - Reno - Yosemite - Death Valley - Vegas - San Diego).
SF - we stayed in Oakland and took the train into town (we had a car by then and hotel parking in the city is not cheap).
Lots of hotels have laundry facilities which makes life easier.
The one way car hire fee can be worth it to save driving all the way back to the start of your trip. Check out the cost versus driving unless you plan a loop to take in all that you want to do.
Book Alcatraz well in advance.
LA - Hollywood is a bit seedy, Griffiths Observatory is good for a visit. The Getty Centre is also good (think you need to book in advance?) but your children might not be so interested. Venice Beach is worth seeing, and Santa Monica Pier is the end of Route 66.
Expect long queues at Universal Studios in summer (although there are signs to tell you wait times). You can pre buy tickets in the UK. Pay more to jump the queues.
Sequoia and Kings Canyon are fantastic - and much quieter than Yosemite (Kings Canyon is like a mini Yosemite).
Hotels.com will give you a free night with every 10 nights you book with them. The chains are reasonable quality and not too expensive - La Quinta, Hampton Inn, Best Western etc.
In more rural areas people eat early (not much open after 8pm).
Hoover Dam is stunning but it will be scorching in the height of summer. Ditto Death Valley.
We did 4 weeks for 4 of us (no expense spared) for £10k in 2015. That included a helicopter flight over the Grand Canyon from Vegas - that was brilliant and was much easier than trying to drive to the rim and then stay. Expensive though at £150 per head (but good fun).