Reading your OP, and if you're going for scenery....
Out of Santa Fe, I'd go over to the Pecos wilderness. Also I'd investigate Bandolier Monument. If you go to Taos, I like the high road, but I can't argue against the low road either. Different types of scenery, both interesting.
Depending on your time, preference, etc. I'd either spend one of your Santa Fe days exploring the Los Alamos area, or drive through there on your way to Chaco.
Head out 285 north from Santa Fe. (Gabriel's is out that way, one of my favorite Santa Fe restaurants, at exit 176) At the Pojoauque (po wok e) Pueblo, exit highway 502 and head up the hill to Los Alamos. Where highway 502 and 4 split, take 502 into downtown Los Alamos. If you have time, 4 is also a pretty drive, and fascinating for the erosional patterns in the ash fall (Bandolier tuff) There's a nice overlook about halfway up the mesa on 502.
As you get to LA, one of the old guard towers is still there. And about 3/10 of a mile past it is a pullout to one of the hiking trails along the mesa edge. Worth a look. In LA are the LA Historical Museum, and the Bradbury Science Museum, both great stops if you have any interest in the Manhattan project.
Leaving LA, head out on highway 501 toward Valles Caldera. You pass through the Los Alamos National Labs. It's a little disconcerting, as you must pass through a guard gate. Just show your driver's license, be calm, and go on through. You'll be driving a public highway through the lab grounds. Were you to attempt to actually go in any of the facilities, you would be subject to much higher levels of screening.
The only legal glitch is if you don't have a license/aren't a citizen, or are carrying firearms. Even then, I don't think it matters, they really don't check closely. But if you get the willies going through security, there is a back road that takes you past the secure area. Look for W (West) Road at Diamond Drive next to the LA Medical Facility. Instead of crossing the bridge into the lab, it drops down into the valley. Kind of a pretty drive, anyway.
Continue with 501 to highway 4, and take 4 into Valles Caldera. Beautiful. From there, you can head on into Cuba on 126 and on to Chaco. There's a pretty hike up the E. Jemez at the Las Conchas trailhead and Jemez Falls is pretty.
When you get to the 4/126 split, you can take 126 to Cuba. Or drop south to Jemez Springs. Stops along there are Battleship Rock and the Soda Dam. On past Jemez, find the little road 485, which inexplicably becomes 376 a few miles down the road. This is a road along the old narrow gauge railroad route to Gilman. It takes you through the Gilman Tunnels and the spectacular Guadalupe Box. You can take 376 over miles of dirt road, north to Fenton Lake, and rejoin highway 4 to Cuba.
