Back in the 90's before we moved to NM, my wife and i would fly into a western city and just rent a car and do a loop through the state. We had no hotel/motel reservations and our only itiniary was to eventually get back to the airport for our flight home. Great, great vacations !!! We did Arizona, New Mexico (twice), Colorado, and Utah on those various trips. Second time thru NM we looked at some houses just for future retirement, and then ended up having a mid-life crisis and moved there on a lark. Fortunately, it worked out very well for us.
Back to your original question.
1. If you want spectacular scenery - the best is Arizona with Grand Canyon, Sedona, and several nice large lakes. Utah is equally spectacular via the national parks in southern Utah - Bryce, Arches, Zion, and Canyonlands are the ones that we visited. There are more.
2. For culture, NM was our favorite with their Native American and Hispanic cultural influences. Acoma, Santa Fe, Albuquerque old city, Taos , and Bandelier all great stops amongst others. Utah, of course, has the Mormon culture. Though NM had the best overall Native American culture, the single best can miss place is in Colorado - Mesa Verde, near quaint Durango. Acoma and Taos are living pueblos, so in that respect they have a layer of interest that even Mesa Verde lacks.
3. For cities with the most to do - there is a ton of stuff in Pheonix, Albuquerque, Denver, and Salt Lake. Smaller quainter cities might include Sedona (our favorite), Durango, Provo, and Colorado Springs.
4. Believe it or not my wife and i also have a sand collection. We have some nice samples from Sedona's red dirt, Great Dunes National Park in Colordo, White Sands in NM, and there is a place in Utah whose name escapes me - too many years.
5. We have also visited Nevada though not on one of our loop the state trips. I am not a Vegas fan, but St George and Hoover Dam/ Lake Mead are highlights.
6. For cuisine - plenty of good place, but nothing is as unique as the New Mexico Chili Peppers. They can be mild, moderate, or hot as you may prefer, but nothing beats a good Green Chile Cheeseberger or breakfast omelettes or Mexican food done right with a Green or Red chile smother !!!
Anyways, no way to take it all in on one trip. Hope you find great highlights and decide to go back multiple times !!!
If i can help with any questions, please feel free to privately message me !!!
PS - Congrats !!!