Some thoughts for you to ponder.
Hello and welcome to the RTA forums !
First of all you are correct, you really won't have time to fit in Yellowstone and Rushmore and have a relaxed road trip.
Secondly. Adapting to driving in the USA is quite quick and easy and you should not have much to concern yourselves with. With a large group it would be wise for everyone to contain their excitement for a little while and let the driver concentrate, with the aid of Co-driver concentrating on the task ahead, allowing you [the driver] too settle in.
Now to your trip. You can not get from Vegas to Grand canyon National park and back the same day and enjoy time at the canyon, even without stopping at the Hoover Dam ! It also adds miles to go back to Vegas before going to LA. You could spend time in Vegas go to the GC and then onto LA, but I wouldn't do that given the choice.
I would drive the whole route starting and ending in LA would look something like this.
Days 1 and 2 in Vegas. Day 3 drive towards Yosemite by crossing Death Valley and stopping around Bishop or Mammoth Lakes CA for the night. Day 4 and 5 into Yosemite on Tioga pass [the spectacular CA120] and spend time exploring before heading to San Fran. Day 6 San Fran.
Now if you want to head to LA same day you will miss one of the most spectacular coast drives in the world around Big Sur on CA1. That will depend on how much time you want to spend in LA. [Personally I would skip the urban sprawl altogether with the time you have and drive the coast to Cambria and then head inland towards the Grand canyon.]
So day 7 along the coast or to LA.[?] Day 8 towards the GC and Day 9 arrive at the GC, hopefully giving you enough time to get back to Vegas for your flight on day 10[?]
You could do the trip in either direction but the one above has the advantage of putting the ocean on your side of the road heading south.
As you are flying in from NY and back to Florida, if at all possible I would add a day or 2 to this road trip and I would highly recommend it, the drives and scenery can be spectacular ! If you can't, I would choose between LA and GC to free up time to explore the other places at a more relaxed pace. If you decide to drop GC *** You can get tours from Vegas but be warned, most go to the West rim which is not part of the National park***** then starting in Vegas and ending in LA would be my option, again driving south by the ocean.
You went to the west rim.
You did not go to the South rim of the National Park if you went to the Skywalk. That is located at the west rim that I mentioned earlier. You want to head to the National park to see the Canyon 'proper' and the most suitably located is the south rim.
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Thanks for the suggestions... As GLC also pointed out, I guess we would take GC the last.
Indeed and exactly the same as I originally suggested. It would be the best way when heading south down the coast and the GC back to Vegas.