Public transport is great from Dublin to Northern Ireland and from Dublin to Galway. You can get to Dingle from Galway,but it will take an entire day of your trip by bus.
Here is what we did on our trip using public transport.
We flew into Dublin and spent two nights there. We took the train to Belfast were we spent 1 night and focused in on the Titanic Experience and the HMS Caroline. The next day was an easy train and bus ride to the Antrim Coast where we spent 3 nights enjoying Giant’s Causeway, Bushmills, Dunluce Castle, and Port Rush.
The next day was the most grueling of the trip as we spent almost the whole day going to Galway by bus via Derry and Sligo. It was quite scenic, but also quite tiring. One night in Galway before taking bus/ferry to the Aran Islands for 2 nights. This was quite easy on public transit and I wonder if you might want to consider trading Dingle for the Aran Islands to give yourself a break. They are quite lovely.
Back to Galway for 2 nights. We took a day trip by bus and boat cruise to Ashford Castle where my sister did the Hawk Walk at the school of falconry. ( she says it was the trip highlight for her). We returned to Galway the same way.
A night in Westport and a final night at Dublin Airport.
There was a great deal of Ireland that we didn’t see, but we tried to enjoy what we had time for and what we could reasonably reach by public transit. I feel that Northern Ireland gives some great options that many people seem to overlook.