The fastest way may be to travel down 287 to Ft Worth and then I-35 to NB. But it is a boring drive.
If you do not mind stopping in every little town between the two then the Ama to Lubbock, Abilene, Coleman, Brady, Fredricksburg, to NB is not too bad.
I have never traveled this route from Amarillo, but Ama to Wichita Falls and then down 281 to NB might not be bad. I have been up and down 281 and it is a pretty good road all the way.
It is a long haul anyway you look at it. Southwest flies out of Amarillo. That would be my choice.