Weather certainly might be an issue in February. I think I would take the San Antonio route, stopping to see the Alamo and take a boat ride on the River. (They have great public transportation so don't worry about parking) Then I would head out on I-10 to Fort Stockton, spend the night and then take Hwy 285 to Carlsbad Caverns. A must do if you haven't seen the caves. My next stop would be Albuqueque where I would visit Old Town and I'd also spend one night at the Alcoma Pueblo Casino and do the Pueblo Tour and another night at Red Rock State Park near Gallup. (maybe a couple of nights so I could do some turquoise shopping in Gallup)
If I had all the time in the world, I would go north from Gallup and check out 4 Corners, and Monument Valley and the west again to the Grand Canyon, but if I had a time crunch, I'd stay on I40, checking out the Petrified Forest before turning off to go to the Grand Canyon. Personally the only way I would visit the Petrified Forest would be if it was on my route as it sounds more interesting than it is. IMOP
Since you don't say where home is, after the Grand Canyon, I would again head north to Bryce Canyon National Park and then up I15 to Salt Lake City making a stop at the Great Salt Lake and then grabbing I 80 for home.
This tour would only be a taste of the southwest, but enough that you'll certainly want to come back for more.
When in Texas try the Barbeque, When in New Mexico try fry bread and Green Chili Hamburgers. Happy travels.