To answer your question, yes, you can bail yourself out of jail. However, most people choose to be bailed out by a loved one using a bail bond company. Either way, the process is the same, whether you or a loved one chooses to bail yourself out.
Bail is seldom denied and for most criminal offenses is a “matter of right.” In order for the judge to deny bail, you must either be charged with aggravated — Utah's only crime that is subject to capital punishment — or be charged with a felony and, e.g., be adjudged to be a flight risk.
If you are taken into custody in Utah and can't afford bail, you will be held in jail until your trial date. This period of awaiting trial is known as “remand detention,” and it can last up to a few months.