Drive. The nearest city with air service is Farmington New Mexico, about 60 miles north, which is served by a commuter airline Mesa that is a partner with United Airlines. The nearest major airport is in Albuquerque about 150 miles southeast. There is no rail or bus service in this remote region.
Older maps of the Chaco area may lead drivers astray owing to a road closure. This quote from the NPS web site link below gives current driving directions: "Hwy 57 from Blanco Trading Post on US 550 is permanently closed at the park's north boundary. Do not take Hwy 57. From US 550, go to mile 112.5 3 miles SE mile of Nageezi and turn onto CR 7900 and CR 7950. Follow signs to the park. Hwy 57 south is open from the park boundary to Hwy 9. There is 20 miles of rough dirt road. Not recommended for RVs." The "not recommended for RVs" comment doesn't do this road justice: if there has been recent precipitation, this is a hairy road to drive that may leave you stuck in a large mud puddle or high-centered on a bypass, in either case about 20 miles from help. Avoid this route at such times, and go around to the eastern connection with the county roads. You'll see two signs at the key turnoff on NM 9, one pointing left to Chaco and the other straight, also pointing to Chaco but giving a longer mileage; go straight in dubious conditions rather than left, and you'll connect to CR 7900.