Can a sixteen year old move out of family home in the state of Colorado?

You can legally move out on your own if your are "emancipated".  Emancipation is a legal process by which minors can attain legal adulthood before reaching the age at which they would normally be considered adults (this is called the “age of majority”; in Colorado it is 19).

However, there are several ways for a child to be emancipated prior to reaching the age of majority.  A child can become emancipated if he or she:

1. Joins the military;
2. Gets married;
3. Lives on his/her own and is self-supporting;
4. By conduct, though you must still prove that the child is self-supporting; or
5. By Order of the Court.

What you need to do now is to talk with a lawyer about all of this.  Don't do anything foolish just to get out now.  What you do now will effect the future.  You want to make sure that your choice is a good one. 

If money is an issue, see if you qualify for Legal Aid or see if they can recommend someone to help you.  Also, check if there is a law school nearby to where you live; they typically run free/low cost clinics.  You can contact the local Bar Association in your county; they may have a list of attorneys who will take your case "pro bono" (for free) or at least for a reduced fee based on your income/circumstances.  Additionally, google childen's support groups in your area, they can usually provide a list of attorneys who can help. Finally, contact your state's Department of Social Services, they may be able to refer you to free legal services.