The statute of limitations depends on the type of contract you made. If it was a properly executed (signed) written agreement then Utah Code 78b-2-309 likely applies which gives you 6 years to file. If the agreement was oral then Utah Code 78b-2-307 likely applies which gives you 4 years. However, if your contract is oral you may have other issues trying to enforce the agreement depending on what your agreement actually was. For example, the statute of frauds or Utah Code 25-5-4 would invalidate a lease for a year or more if it is not in writing.
I hope this helps. To really know for sure which statute of limitations applies and whether your agreement would be enforceable I'd need to know more about the specifics of your agreement.
Regardless of whether your agreement is oral or written you should proceed with suit. The longer you wait, the harder it gets to collect. You can get most of the money - up to $10k by going to small claims court. It is cheaper than hiring an attorney. However, an attorney may help move things faster.
This is not legal advice. Even if it were, fee legal advice is worth what you pay for it. The facts of every case are different and should be addressed by a competent attorney who has all of the facts.