I assume you will generally be following the Rio Grande until you get very near Santa Fe. Your route will approximate the old Spanish El Camino Real de Tierra Adentro (Royal Road of the Interior), which has a facinating history. There is a lot of information about that historic route in the internet. Here's one site http://www.nps.gov/elca/index.htm
I did a bike tour from Las Cruces to Santa Fe about 5 years go. Interstate Highway 25 follows the Rio Grande to Santa Fe. It is legal (and safe due to wide shoulders) to ride on the interstate, but noisy and not real pleasant. Most of the route there are paved highways parallel and not far from the interstate, some east and some west. These roads are much more pleasant riding. A detailed New Mexico map will show those roads. If you're not camping, towns with motels are spaced near enough for reasonable daily rides. There is very little civilization between Truth or Consequences and Socorro, so load up with water and snacks for that stretch.
New Mexico Highway 28 north from El Paso goes through La Mesilla, an interesting and attractive little town just south of Las Cruces. Of course, you will find many, many good Mexican food restaurants along the entire route.
Between a bit south of Albuquerque and Santa Fe are several Indian pueblos (towns) that are interesting to visit. In Albuquerque, a 15-mile bike trail runs along the Rio Grande.
As others have said, it will be hot; along with water, sunscreen is a must. Long sleeved shirts may be advisable to avoid sunscreen. Just for information: Las Cruces elevation is 3,900 feet, Santa Fe elevation is 7,200 feet.
Have a great ride.