It will take about three hours to drive from Yellowstone to the Tetons, depending on where you are staying, then 8 to 9 hours to Moab, then 6 to 7 hours to Vegas. So conservatively say 3 out of your 13 days, that leaves 10 days. I've been to Yellowstone and the Tetons multiple times and also to Arches, Canyon Lands and Vegas 3 times.
If you can, maybe stay at two different locations in Yellowstone to cut down the driving. I really liked the Lake Hotel and Lodge and then the Old Faithful area. Say 2 days at each location. My favorites locations in YS are Lamar Valley, Grand Canyon of YS, Mammoth (the Beaver Ponds hike is my favorite) and the Old Faithful Geyser Basin. Don't miss the Grand Prismatic Pool.
Then two days at the Tetons. It's a fairly small park and you can cover it in two days if you don't do much hiking. I'd suggest a day at Jenny Lake and hiking up to Inspiration Point and on your way, on the south side of the lake are the Moose Ponds where one can often see Moose. I saw 5 there at the same time in the fall of 2022. I also suggest driving Moose Wilson Road, there are often black bear and moose along the road. Then on the east side of the park, across the highway (close to Moose) is Morman Row where one sees the classic view of the Tetons and the old barns.
Then two days in Moab. You can cover Arches in one day if you don't do any long hikes. Get there early to hike to Delicate Arch, it will be very warm in the summer. Most of the notable arches are just a short walk from the park road.
Canyonlands can also be done in one day if you just drive through and look at the overlooks and the short hike sites. If you have time on your way out, stop by Dead Horse point, its a very beautiful area.
It takes a little more time to drive, but my suggestion is to go north from Moab to Interstate 70 and then take highway 24 to highway 12 for your trip to Vegas. You will pass through Capitol Reef National Park and even if you don't stop, there are some incredible sights right on the highway. Then route 12 goes through Grand Staircase Escalante National Monument and that route is an Utah Scenic Byway, a very nice drive. And if you start really early you could stop at Bryce Canyon NP for a couple of hours, it's enough to walk the main part of the rim trail. Lots to see in Utah.
That leave 2 days for Vegas, one day to drive to Valley of Fire State Park (yes it's worth it) and another day to walk the Vegas strip and see the sights. Maybe catch a show at night. I have lots of photos on my web page if you want to see each of the areas.
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