How do you get from galway Ireland to dingle Ireland on public transportation?

The best way is probably by bus. You may need to make at least two journeys. From Galway you could go to Limerick for example and then on towards Dingle. See the related link below for the main bus company, from where you can look at your options.

You need to make changes in Limerick and Tralee, but you can ask the driver on the first bus for a through ticket to Dingle which will be cheaper than buying 3 separate tickets.