Welcome aboard OCDO - generally problems are few in NM.
New Mexico is one of our “Gold Star” open carry states. There is complete state preemption of all firearms laws, open carry is increasingly common and law enforcement is well educated as to its legality. In addition, New Mexico has an extended domain law which effectively makes your vehicle an extention of your home. So you may carry openly or concealed in your vehicle.
There is no "registration" required of your handgun.
Carry whatever flavor of ammo you prefer - a good choice is often what local LEOs carry. Though problems are few, a good digital recorder can be a welcome asset.