Well, you can knock off 4 states in a half hour or less - UT, CO, NM, AZ - Four Corners.
On the way from Texas take US 285 from Pecos, TX all the way to Pojoaque, NM (north of Santa Fe), then NM 502 to NM 4 just east of Los Alamos, then west on NM to Bandelier National Monument and Valles Caldera National Preserve. Continue on NM 4 to La Cueva, NM, then west on NM 126 to Cuba, NM. In Cuba, take US 550 north to Bloomfield, NM, then west on US 64 through Shiprock, NM to Teec Nos Pos, AZ. In Teec Nos Pos, take US 160 northeast past the Four Corners. After passing the Four Corners on US 160, you'll enter CO. In a few miles, turn left (west) on CO 41, which becomes UT 262 in a few miles. UT 262 takes you past Hovenweep National Monument on the way to Blanding, UT. From there, you have a lot of choices to see the scenic southern Utah canyon country - UT 95 through Hanksville and Capitol Reef National Park or US 191 through Moab and Arches National Park and Canyonlands National Park.