Gala & Steve
Welcome to the club
There are so many great National Parks and Forests to visit. My first suggestion would be start in Texas and got the Big Bend NP, best seen in late fall through early spring, because of the heat If you are up to a challenging hike got to Guadalupe NP (also in Texas) and hike to the top. Carlsbad Caverns is only 35 miles from Guadalupe NP.
In that end of Texas it is worth visiting the Davis Mountains State Park, McDonald Observatory and Balmorhea SP.
The place suggested in Utah are wonderful along with Zion, Bryce, Grand Escalante, Cedar Breaks, etc Don't miss going to the Grand Canyon (both the south and north rims) we prefer the north rim. We made a loop one summer through the Grand Canyon, then down to Sedona, back up to the North Rim and all of the National Parks in southern Utah.
Another summer we made a loop from Texas going to Grand Tetons NP, Yellowstone NP, Devils Tower NP, Mount Rushmore, Crazy Horse, Custer State Park, Black Hill, Badlands NP and Sturgis, SD then on home.
Enjoy your adventures