Jeremy,
Unless you really enjoy hot, HOT, weather, I would go in April. I have made three trips to the American Southwest, all in April, and the weather was great, trees were just leafing out, and there were fewer tourists around than there are generally in August.
Of course, while we all know that Blas doesn't know a thing about anything important ( -- I have to tease him), I agree with him on White Sands, which was first on our New Mexico trip last year. There are two problems with WS, however: The military missile range conducts tests a couple of days each week, and on test days the road to and past White Sands is closed most of the day. We didn't know that, but lucked out as a sand storm canceled the tests and the road stayed open (so I got some nice photos inside the sand storm, plus several big dust spots on my sensor, even though I never changed the lens even once.) The other problem with White Sands is that it is pretty far from your other destinations, especially if you have only ten days.
If you can possibly stretch it to three weeks, you can do a lot more. In 2008 my wife and I did three weeks just in New Mexico, and we barely scratched the surface. My favorites in New Mexico included White Sands, Bosque del Apache (great for wildlife and birds), and Chaco Canyon National Park (one of the best Anasazi sites around).
This year we did parts of Southern Utah, including Zion, Bryce, and Capitol Reef National Parks and parts of Escalante/Grand Staircase National Monument. That filled three weeks easily, and we never got to Antelope Canyon or Monument Valley or the other Utah national parks. We plan to get to Arches and Canyonlands on a separate trip.
If you want Grand Canyon, Antelope Canyon, Bryce, and Monument Valley, you will need to visit the North Rim of Grand Canyon instead of the South Rim, because driving around will take too many of your days. The North Rim, as I understand it, requires a fairly long drive on a fairly ratty road, but could be well worth it (I have only done the South Rim, many years ago).
There is so much bizarre and fascinating landscape in that part of the world that I'm sure you will find lots to explore. Because the distances are so great, I suggest picking a relatively smallish area. For southern Utah, we flew into Las Vegas and used a renatl car for the trip.
You can see a few of my Utah photos on my website, in both the B&W and Color galleries and Calendars. I just realized I don't have a New Mexico gallery up, so I'll have to remedy that soon.
Cheers,
Eric