Fastest is most likely I-45 to Dallas, US-287 to Amarillo, then I-40. It is the shortest (1300 miles) making use of good highways. This depends on what time you hit the Dallas area and how messy traffic is around Denton. Most of 287 will be lightly traveled two and four lane rural highway with speed limits at 60 to 70 mph.
I-10 to Phoenix, then I-17 north is 120 miles longer, and about 2-3 hours more time, but you spend all your time on Interstates so you feel you are going faster. Traffic areas that can slow you down depending on timing include San Antonio, El Paso (to Los Cruces) and through and around Phoenix.
Going I-10 to El Paso then I-25 up to I-40 is only seven miles longer than going through Phoenix, no real difference in time.
For me it would depend on my departure point in Houston. If leaving from Sugarland I would head out I-10. If leaving from the Woodlands I would go up I-45. Getting across Houston from one to the other would use up any difference in travel time on the next 1200 miles of the trip.
Tom Test
Itasca Spirit 29B