Colorado unlike where you are coming from has lots of public land. BLM, Forest Service and State trust land. So there is LOTS of places to ride. But you don't find a lot of adverised Horse camps like back east. Most of us just pull off into the trees and ride.
Riding in the west is no different than riding anyhere else from a safety standpoint. It is big country and you can go miles before crossing a road. I would learn how to read a map. Maybe a GPS. Riding in the mountains can put you in a changing tempereture zone. Where it can be 70 and comfortable at noon and 30 and snowing at 2pm. So always layer clothing and be prepared.
You might check out FourConersTrails.com for helping finding trails
There is also a strong group that does Competive Trail Ridig CTR in that area. Chck out www.NATRC.org
and look a their calendar of ride.