No laws against possessing a firearm on a National Forest, unless it's prohibited by State laws. Check the New Mexico state laws about transporting firearms in vehicles, loaded-unloaded, carrying out of hunting season, that sort of thing. If you're in compliance with State law, you're in general compliance with federal regulations on NFs. Of course that also includes the state laws about brandishing a firearm, alcohol while armed, etc., etc., so be prudent at all times.
On National Forest land, you may not shoot in or near any campsite or occupied area, over or across any road, trail, body of water, or without a safe backstop. Trees are not safe backstops, they are valuable and don't shoot them up. Don't shoot in an unsafe manner. The federal regulations are not about possessing a firearm, but regulating the use on NF land.
You can contact the Santa Fe NF on the web and inquire directly about areas that may be closed to shooting, and the general regulations. There is a law enforcement officer there who will be glad to keep you on the straight and narrow.
It's a beautiful Forest, you'll enjoy it a lot.