South East
If you are starting in the SE and going to work your way up, You must go through White Sands.
You don't need to spend the whole day, but you should at least drive through.
The Trinity site would be a must see. Not sure if you can just go any time.
Albuquerque
If you are a fan of Breaking Bad, you can drive around Abq and see some of the more recognizable landmarks from the show.
I've always been a fan of the Atomic Museum (also in the show). It has some great history there.
Balloon fiesta museum
You could ride the Tram
Santa Fe
The Plaza in Santa Fe should also be on the list.
Loretto Chapel
Drive by the Governors mansion while sticking your hand out the window and show our governor only you middle finger.
Taos
Rio Grande Gorge bridge outside of Taos
Taos Plaza
El Santuario de Chimayo
For a quiet relaxing couple of days, you could stay in Red River. Nothing special to see, but a decent place to chill out for a few days.
If you make your way toward Chama, You could ride the narrow gauge railroad
I'm sure more will come to me as soon as i click submit