What route would you take to get to Amarillo? You could take I-45 North to I-35 North into OK, then west on I-40 which would bring you back into the Texas panhandle and into Amarillo.
If you're thinking of going North on I-45 into Dallas and jetting Northwest toward Amarillo via highways (287?), I'd avoid it. I traveled from Amarillo to Dallas on Highway 287 last year and there were way too many small towns where you had to slow down and traffic lights, too.
As for traveling through the other states,