How do I find out when I became a US citizen?

By looking on your Certificate of Naturalization (Form N-550), which you and others naturalized the same time as you all received as part of your naturalization ceremony held at a courthouse and were told to always keep, as well as told to immediately record at your local county records office in that same building that day so that if you were to ever lose it, there'd be a public record of it that you could get a certified copy of by simply contacting that county records office.