I would give, not take day's.
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We have two weeks set aside for this (with the ability to give or take a few extra days).
If you want to do some sight seeing along the way I certainly wouldn't give up any days, but I would try to add some. You have around 9 days of dedicated driving with time for the basic stops for food, rest and a stretch of the legs etc, and with 3 days in Chicago that only leaves a couple of days to look around Memphis, NOLA and Albuquerque as it stands, never mind other things.
There are a thousand and one possible attractions between LA and Grand Junction alone, which is also just one of many 'places to stop going in that direction'. ;-) This is why we don't do 'best' around here because each trip is unique.
In the green tool bar above you will find the RTA Map Centre by putting your cursor on the 'Maps' link, you can put in your route[s] and find hand picked attractions by RTA contributors from first hand experiences. Once you have researched and discovered what routes are best for you, we can help to fill in the blanks and 'Fine tune' what you have, but unless you find more time you won't have much time to spare. There are many great National Parks and driving roads through Utah and Colorado which could take a fortnight just to scratch the surface. Moab would be an option for an overnight stop after Vegas to include a quick visit to Arches or Canyonlands NP, near Fruita there is the rim drive of the Colorado National Monument just off I70 and a little further along near Dillon there is a spectacular drive over the Continental divide at Loveland Pass, but even 3 detours like this will eat away the hours.
Enjoy the planning !