Best Route from NorthEast to Southwest?
Can't speak about Arizona to LA, but from Tucson to upstate NY, we went on I10 and I20 to Ft Worth, on I30 and I40 to Nashville, on I65 to Louisville, on I71 and I90 to Upstate NY. A nice trip, but perhaps a bit longer. We went this way only because we had friends in Ft Worth.
Coming back, we used to go on I90 to I71 to I70 to St Louis, I44 to OK city, I40 to Albuquerque, I25 to Las Cruces and I10 back to Tucson (There is a shortcut from I25 to I10 at Deming on back roads). You could follow I10 back up to LA.
But the 2 sections of Toll road in OK are very expensive (based on number of axels), so last time we continued on I70 to Kansas and took I35 down to OK city. Still had to pay toll in Kansas, but much cheaper than the OK rate. Probably will continue using this route from now on.
These directions do not include short sections of roads between the major highways, or bypasses around big cities (which we generally give preferance to).
Note that you can save as much as 240 miles by not going down to Tucson and back up to LA, but you will go through a lot more steep grades. This way is alleged to only have one (between Blythe and San Bernadino).