If you do a Google search on "dog friendly camping in Colorado", you'll get a lot of valuable information on what to expect at various locations around the state. As a proud Weimie owner, we've used these resources when making plans.
Additionally,
H E R E is a link to information on where pets are allowed in the multiple National Parks and Monuments in Colorado. Notably absent from this list is Mesa Verde National Park. It's restrictions are similar to RMNP, adding in archeological sites as prohibited.
If you could provide more information on a particular area of the state you're interested in as well as how much time you've got, would be helpful in our providing more substantial answers. Some great suggestions so far!