I, as the lawyer, look at this question in two ways. Do you mean where in the geographical sense or do you mean where in the physical sense?
The physical sense is at the actual district court house. A district court is different from a juvenile court, justice court, appellate court, and even the Supreme Court. You can Google “Utah District Courts” or go to http://www.utcourts.gov to find the one nearest to you. Even if you end up in the wrong place, someone at the wrong court will tell you where the right court is. I have bailiffs do this, court clerks do this, and yes — even lawyers do this (without charge of course). Do not fear you cannot file the papers at the wrong type of court.
Once you have found the right district court, you walk in with your paperwork (leave all weapons and metal in your vehicle or at home), ask where the clerk’s office is from one of the police officers with the job of working at a court house, and the court clerk will kindly (or not so kindly depending on the person and the day as well as your level of competence) help you file the papers. You pay the filing fee ($310) plus an $8 vital statistics fee and you have just started your divorce! If you cannot afford the fee, there is a document you can file to ask the court to waive the filing fee. The forms can be obtained on line as well as at the court.
If you are being abused or have a fear of being abused, this can be a terrifying yet exhilarating step on your way to freedom. It is a brave and bold moment that starts a recovery to getting back to taking care of you.
In the geographical sense, you file the divorce paperwork in the County where you live. Of course that means that you have lived there for longer than three months and if you have children they also have lived there for three months as well. I get calls from spouses who have fled their husbands who live in another state and usually are returning to Utah to live with their parents. They still have to wait the three months before filing. I guess the point behind this is to keep people from fleeing to the State of Utah for a divorce!
• you file for divorce at a district court house
• you file your divorce paperwork with the district court clerk
• you file your divorce paperwork in the County where you live
• you pay the filing fee when you file your divorce paperwork