I often camp by Diablo Canyon, which has wonderful sunsets and decent off trail hiking. There's even a cave for those who know where to find it.
I've also camped down by the river for the access to water for splashing about and cooking, but the area is popular with Mexican families and sometimes attracts a party crowd.
There are plenty of more remote spots off the main gravel road that would be even quieter. The terrain is juniper and oak, with cottonwoods by the river.
Temperatures run hot in the summer and freezing in winter, although it is the best choice for winter camping near Santa Fe. Summers the wife and I most often drive up Artist Road and camp at Big Tesuque or the skiiiway at the top, but we did choose Buckman this summer when thunderstorms pounded the mountains.
It does take time to drive the gravel road, and there's not much for ordinary folks around Las Campanas, the wealthy housing division between the camping and town. You might get to Airport Road and Cerrillos under an hour, but getting to the Plaza will take longer.
Cell service is spotty.