Dennis, if driving in western Ireland isn’t under consideration, I’m guessing that bicycling isn’t, either. Fourteen years-ago, age 61, my husband and I made our way down the coast, south from Galway, over to the Aran Islands, and down to Dingle. There were lot of hills, and some rain, too.
But we also used the Bus Eireann bus system twice, from Tralee to Dingle and from Dingle to Limerick. The buses have a running Irish Setter along the side, instead of a greyhound. Trains do serve some cities, including Ennis, but as you said, the rail network is far less extensive than in countries farther south.
We also used area taxis occasionally, including getting from our lodging to Lisdoonvarna and from other lodging into Dingle town. Our B&B’s had information about options, but then you’d need to get from one B&B to the next one. I don’t know what private driving services might exist, for covering some distance a single time.
If you did want to visit Inishmore or other Aran Islands, there are ferries, which we used when bringing those bikes with us, but on a subsequent trip (with rental car), we discovered the Aer Arann Islands airline, with small planes. Faster and more weather/waves resistant than the ferry boats, the planes are the way to go.