East Meadow, New York
will be leaving saturday and would like time to leave and hotels to stay in
A lot of it depends on how many days you want to drive, and how much time you have to explore interesting stops along the way. With a simple 2-day road trip, you can basically leave at 9am each day and do a full day's driving with a few relatively short stops. But it might make more sense to take 3 days so you can spend more time hiking in the parks. Plus if you're driving through Vegas, who knows how long you might end up there...
If you're on a really casual pace, I might even pick a hotel in Vegas for the first stop if you don't go there a lot. If you're into camping, the national parks would have a lot of great options to stay overnight for cheap.
Hopefully other people can give you more specific advice from their experience, good luck with your trip!
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