Simple answer is: it depends. Most divorce cases are settled, either through the attorneys, or in mediation. But, both of those options require that you and your spouse agree on all the issues in your divorce. If you do not agree, then yes, you will need to go to court for a trial.
No matter what, understand that a divorce is a legal action – there will be a court case filed. In Utah, family law cases are privileged, so they are not “public” record.
You may also want to go to court while you and your spouse are working things out. The reason being that you may want to obtain temporary orders for custody; parent-time; child support; use of the family home; and to determine who pays what. Any such temporary orders would control your divorce case until such time as the final decree is entered.
Besides having various issues settled, on a temporary basis, going to court for temporary orders can be useful if the other side decides to use self-help to gain access to the children, simple takes the children, or acts in inappropriate ways violating those temporary orders. In that case, you can return to the court and ask that the offending party be admonished, or punished, for the inappropriate actions, including the possibility of being awarded your attorney fees for having to bring the motion.
So, no matter what, you will have to have a court case in order to actually get divorced in Utah, and actually get your Decree of Divorce. Whether you actually go to court depends on your needs, and your situation at the time.