Plane
The nearest major airports are in Phoenix, Arizona and Albuquerque, New Mexico. Once you get to the airport, you may want to rent a car and drive to the park.
Private Vehicle
Driving from Phoenix you have several choices:
1) travel Interstates 17 North and 40 East, passing through Flagstaff (259 miles), or
2) travel Highways 87 North to Payson, 260 East to Heber, 377 North to Holbrook, and 180 South to the park (215 miles).
Driving from Albuquerque travel 204 miles west on Interstate 40 to Exit 311.