San Antonio is a very pretty city, but it is a small city. You can see most of the major sights in two or three days. One week should be more than enough time to visit San Antonio.
If you want to drive from San Antonio to Santa Fe, you must drive for about 15 hours. It will take you at least two days of hard driving to go there. Texas is a big state, and there are not many interesting sites in West Texas. You will have many hours of boring driving across some flat plains.
I would consider taking an airplane to Albuquerque and renting a car in that city. There are some interesting things to see in New Mexico. The Pueblo communities like Acoma, the sky peublo, a few hours west of Albuquerque. White Sands, Carlsbad Caverns, and Rosswell, home of the space aliens, are all within a days drive from Albuquerque.
Santa Fe is just a few hours north of Albuquerque, and the beautiful mountain resort of Taos is only a few hours further. If you are eager for even more adventure, you can continue driving north for another half day and visit southern Colorado.