Without question, a car is the best way both to reach and explore the region, as public transportation options are few and won't get you out to see the parks, attractions, and smaller towns, which are many miles from one another. This is a beautiful region for road-tripping, with little traffic—just remember to always set out with a full tank of gas, as services are few and far between.
Most visitors enter the region via Interstate 25, either driving north from Albuquerque and Santa Fe, where the first major stop is Las Vegas, or driving south from Colorado, where you'll first reach Raton. Less-traveled but still scenic (in a wide-open-spaces kind of way) routes through the region include U.S. 87 up from Amarillo, Texas, to Clayton, and U.S. 64 from the Oklahoma panhandle and southwestern Kansas to Clayton. There are also a few stunningly picturesque routes into the area from the west, over the Sangre de Cristo range: try U.S. 64 from Taos into Cimarron, or NM 518, which you can take from the High Road to Taos (over the mountains from tiny Peñasco to Mora and eventually Las Vegas. These mountain passes see a lot of snow in winter, so check conditions. Finally, a quick and pretty shortcut, if you approach from Texas via Interstate 40, is to head northwest on U.S. 84 from near Santa Rosa to Las Vegas.