Hmmmm...really big state lots of miles of driving if you really want to see things.
BTW - I keep all of my cameras handy wherever I go, locking things in the trunk is ridiculous. I've traveled all over the state for over 35 years and have taken photos off of 380 (north end of WSMR), down 54 (east side of WSMR) etc. The range patrols never stop you or even question you because you're on the public side of the fence - and there's nothing to see anyway until you get deep into the test ranges.
If you're coming from Alabama, I'd take I-20 and enter NM at the south end. Now the east side of the state is F-L-A-T. In fact, I think if you look in the dictionary under "flat" it'll say "see west Texas or Eastern New Mexico."
Eastern NM is almost Texas. The southeastern part of the state smells like crude oil (or money depending upon your point-of-view)- and is extremely picturesque with clumps of pump jacks strewn about and forests of wooden power poles supplying electricity to the pumps.
The middle of the east side is heavy duty tilled agriculture, and the northern part of the eastern side is range land.
As you move towards the middle of the state, you run into a ranges of mountains that are oriented north / south and extend all the way from the Texas border to the Colorado border.
Since you're at the south end of the state - I'd get off the interstate at hwy 285, take that north to Orla, and get on the secondary road that takes you over to the Guadalupe Mountains. Lots of interesting things to see along the roadsides on your way to Carlsbad. If you like caverns - okay, do the cavern tour, see the bat flight etc. But, that will take you one day.
Go to Alamagordo via Artesia (mmmm...more great oil smell from the refinery).... I find the road from Artesia to Alamagordo really interesting with lots of weird things to photograph. White Sands is really cool IF you go all the way to the very back, get out of the vehicle and hike into the dunes.
If you go north into the dunes they are medium size and rolling with lots of playas between them. If you go west, they become huge, and on the order of 80 - 120 feet high. There are several problems associated with the dunes. First, everything looks the same and it's easy to get lost. A GPS unit is invaluable. Set your entry point so that you can get back to your car.
Second, is getting access to the dunes when there is good light. In the good old days when it was only a National Monument, it was open 24 hours per day and you could just go in any time.
Now, it has hours. During a full moon they will let you stay out on the dunes until 10:00 pm. Otherwise it's one hour after sunset. If you want to catch the early morning light, you'll have to make arrangements with the staff to meet you at the gate and let you in. Do this as far in advance as possible. There is one campground within the Park, that you hike into. You have to make "reservations" in advance. If that sounds interesting to you - then make your reservations ASAP.
You could spend your entire vacation at White Sands - really, it's that interesting. At the very least, you'll end up spending one day there if you hike into the dunes and see the real dune areas. If you just drive through on the road - you won't see much except the smaller dunes and a bunch of chidren sliding down the dunes. If you get there mid-week there are far less touristas than weekends. If you don't want to spend an entire day there - skip the dunes completely.
It's nearly 300 miles from Alamagordo to Santa Fe. Frankly, Santa Fe ain't what it used to be - I only go there under forced protest or to see the art museum. But, if you like trinketts and touristy thingys - then go ahead if you must.
Personally, I'd head over to the western side of the state and look at the Silver City area with the Gila Mountains. Whip up the western side of the state on 180 along the mountains to 12; and then 12 to Reserve, Aragon, Horse Springs and Datil; then 60 to Socorro. South from Socorro to San Antonio - a quick stop at the Owl Bar for a chile cheesburger and plate of chile fries (mmmmmmmm...). Then on to Carrizozo on Hwy 380. If you want to see lava, there is a lava flow (the Malpais) along 380 at the Valley of Fires state park, just west of Carrizozo.
Just east of Carrizozo at Capitan is hwy 246 that will take you in a loop through some really cool range lands and you'll end up at Roswell. Or, stay on 380 and go to Fort Stanton, and Lincoln and do the Billy the Kid stuff (or not). Once again, at Roswell, the word flat comes into play - but, this is the end of your NM part of the vacation.
Head across west Texas on 380 to 84, then down to I-20 and back to lovely Alabama.
That's the point. I counted and I think between us both there will be six cameras. (We're both photographers, but most of the cameras are mine 
. That said, if you go up via the panhandle, might plan a stop at Palo Duro canyon...good camp ground, just south of Amarillo, from there I-40 into NM and then North on almost any road to I-25 makes a really nice drive with lots to see...think I have driven them all at least once.
