Mengxu's suggestions make good sense
1) definitely need 9 days. We did a very similar circle in 8 days w/o going to Bryce and just drive thru Zion. (but we spent 2 days in Scottsdale/Phoenix/Sedona)
2) could be a good idea to combine Yellowstone, another 8 to 9 days - Yellowstone itself 4 days (we used 5 days and started from JAC, not SLC - saved at least 6 hours driving).
2) could be a good idea to combine Yellowstone, another 8 to 9 days - Yellowstone itself 4 days (we used 5 days and started from JAC, not SLC - saved at least 6 hours driving).