It's so exciting to hear about your Irish roots! Ancestry does indeed offer a DNA kit. Have you tried the paper and document route, by chance?
Is it possible to try and find your parents birth certificates at your local registry office and archives? This way you can find information from their parents, grandparents etc. until you reach your ancestor who wasn't born in the United States but in Ireland instead. :)
There are some super handy resources and tips in this Helpful Research Thread
, and if you have any other questions, please feel free to get in touch!