There are many secondary highways you could take but the direct route is:
From Lyons, KS via 54/64/87 to Amarillo, TX. Then I-27 south to Lubbock; 87/349 to I-20; 329 to Fort Stockton; I-10 west to Balmorhea State Park - nice; then to Fort Davis State Park - nice and take a tour of the observatory; on to Alpine and 90 to Marathon then 385 into Big Bend.
We always stay at Rio Grande Village in the dry camping area - pleasant with some trees, nice walking and mountain views. Right next door is a concessioner RV park with full hookups but you're parked all in a row. This is spring break time so have reservations.
This is approx. 800 miles.
Leaving Big Bend, take 385 to Marathon then 90 to I-10 at Van Horn and continue west on I-10. There are many RV parks at Deming, NM if you're ready to stop at that time. From Las Cruces, NM to Benson, AZ is a long stretch of open desert so either stop at Deming or go on to Benson, AZ.
In Benson, AZ Kartchner Caverns State Park is a very nice campground and you can take the awesome tour of the caverns. Also in the area is Tombstone (try to go on the weekend as it's more festive) and Bisbee. In Bisbee, the mine tour is very well done by former miners. You don a yellow slicker, hard hat with a light and get on a mine car for the tour.
Lots to do 40 mi. west at Tucson and doable from Benson as day trips. Definitely take in the Desert Museum (outdoor critters) and drive through Saguaro Nat'l Park to see some big cactus.
Then pick up 86 in Tucson to take you west to Why, AZ and Organ Pipe Cactus.
This is approx. 730 mi. from Big Bend.
Have a great trip!
