What are Fathers' Rights in Divorce or Paternity in Utah? The Utah Supreme Court says that, if all things are equal, there is no preference for the mother. But from my experience, the courts often seem to not make all things equal. So, you have more of an uphill battle if you want to fight for anything like equal custody. Custody and Parent-time
A lot will depend on the facts of the case; you will need to document details of problems or abuses. And, you cannot use the child directly - never put the child in the middle by asking what the child wants, who the child wants to live with, or anything like that.
In general, the standards and procedures for determining custody apply: If there is no agreement between you and Mom, then the matter will go to court, and you probably should have a custody evaluator appointed to determine what the children's best interests are, and what, if any, problems the other parent presents to the children.
Before you even start to thing about an agreement, or mediation, you need an experienced family lawyer working with you to explain what joint custody actually means, and help you avoid the pitfalls of just going with legal joint custody - which could end up giving you no rights whatsoever about how to raise the child, and leaving you with the standard weekend visitation.
However, if "mom" was married at the time of conception and birth, and you are not the husband, you may have no rights. Go to http://www.hartwig-law.com/blog/2016/08/paternity-actions-when-a-child-is-born-to-a-mother-who-is-married-to-someone-else.shtml