You will need to narrow it down.
Hello and welcome to the RTA forums !
With 3 weeks you have plenty of time to wander and see a number of attractions out of the thousands of possibilities, so many in fact it's hard to narrow them down without you first getting some ideas jotted down. Have you anything in mind ? Personally speaking I would look at heading north through New Mexico and into Colorado through to Southern Utah and then drop down into Arizona. Just a few of the highlights that come to mind could include Carlsbad caverns, Santa Fe, Taos Pueblo, Great sand dunes NP, Black canyon of the Gunnison, the 'Million dollar Highway through Ouray and Silverton, Mesa Verde NP, and in Utah, Arches, Canyonlands, Capitol Reef, Bryce and Zion NP's. Monument valley and the Grand canyon would be top of most lists in AZ and some old route 66 towns and Hoover dam would be popular. The Mojave preserve and Joshua tree NP might appeal or you could look at Las Vegas and Death valley NP. You would even have the option of then going through Yosemite NP and then drive down the spectacular coast road through Big Sur to LA. I am not suggesting you try and see everything and there is a lot more besides, but take a look at your options and when you have an outlined plan we can help to 'Fine tune' it and make suggestions along your route.
So much to choose from - where to start?
Although there is no end of scenic and mountain routes in CO, here's my tuppence worth.
Head straight north out of Taos and cross the border to San Luis. Here you will find a magnificent pathway up a hill, interpreting the Stations of the Cross in life size bronze statues. The cross at the end of which dominates the town. It is truly magnificent.
Up to Alamosa from there to the Great Sand Dunes NP. Allow some time to get to the top of the highest dune, which of course changes with the wind.
You could pick up US-160 west over Wolf Pass, through Pagosa - great hot springs - and onto Durango. From there head up to US-550 to Ridgecrest. There are several national parks in the area. You could take CO-145 back down to Cortez.
Or you could pick up US-50 and head west on it. Just so many options.
You can take any of the other scenic routes. Be sure to pick up a Colorado issued State map at a visitor or Welcome centre, and you will see all these marked on it.... and ever so many more.
Then again, you may want to head to Rocky Mountains NP and drive the Trail Ridge Road (if it is open).
The choice is yours.
Lifey