As you are going to Santa Fe, not southern NM, I have a Florida route I've been using regularly since the 1990s.
I-10 to Mobile. US-98 to Hattiesburg. US-49 to Yazoo City, where I take 49W to Indianola (this is part of the Mississippi Blues Trail). US-82 across to Lake City, Arkansas, then US-65/I-530 across Arkansas Delta Country to Little Rock.
From Little Rock, I-40 all the way west to New Mexico. From El Reno to Tucumcari you are basically following the final routing of Route 66, many historic sites, including revitalized 1940s and 50s tourist traps.
From Tucumcari there are several options. You could break away from I-40 early and try following the original Route 66 through Las Vegas (it is on state highways until it reaches LV, then you are dumped onto I-25). You could take US-64 to I-25, west of Santa Rosa, which is a later era Route 66 visitor site. The best driving is probably I-40 to US-285. While you won't see Las Vegas, you also won't get into as much mountain driving. Some of that is necessary, though, as your destination is in the Rocky Mountains.
To get around in New Mexico you'll likely be using numbered US highways and some state highways. A lot of the most interesting places in the West are not served by the superhighways that connect major cities.
You don't need to go near any of Texas' major cities, or through Louisiana at all, for a northern New Mexico destination. Biggest city in Texas on I-40 is Amarillo. New Orleans, Houston, San Antonio, Dallas/Ft Worth need only be approached if your plan is to visit them.
