The petitioner must file for custody with the appropriate district court. For information about how to file documents, see the Filing Procedures web page.There are two parts to custody: legal custody and physical custody. Legal custody is about who has the right to make important decisions about the children. Going through a child custody dispute in Salt Lake City, UT? Call Pearson Butler about your case today! Learn more about how to to serve civil papers relating to judicial actions filed with the court in Salt Lake County. D.C. Bar Pro Bono Center. 724. Subscribe. There are two ways to modify a child support order in Utah. First, you can file a Motion to Modify Child Support.