A little late to the game here but I've been out playing too much.
Places to see:
Chaco Culture. Not near Santa Fe or Taos, but definitely something to put on one's life bucket list. Way out in the middle of nowhere. There is a CG there, but the dirt road discourages most from dragging their units out there. We hope to go back there again this summer.
El Morro NM. Also not near SF, but a stop that totally surprised me. For both of these, we stayed at McGaffey NFS CG SE of Gallup and did day trips. The McGaffey CG has six or seven sites with electricity and sewer, kind of unusual for NFS CGs.
As previously mentioned, Bandelier NM has a CG but no hookups. It is in the desert but higher up in elevation. Further up NM 4 there are more NFS CG's that are more in the trees. A bit of a commute down to the SF/Taos area, but nice setting.
NE of Santa Fe up NM 475 there are several NFS CGs plus Hyde Memorial SP. Good access down to SF.
There are several NFS CG's above Taos. Never been past them so can't comment on them. Do check into whether any were impacted by the fires last summer.
So much to see and do in Santa Fe. One thing we found is that if you visit on the weekend, you can park for free in the big lot across Santa Fe Trail from the capital. (Hey, I'm frugal not cheap!).
I would point out that they are predicting a wetter than normal winter for NM this year (whew!), so snow might be an issue at higher elevations until you get into May. Note that many NFS CG's don't open until mid-May.