Irish rail seems to not be an easy option to travel to the west and
then rent a car.
If you just want to get from Dublin to someplace in the west or southwest (Limerick, Galway, or maybe Cork), the train is a perfectly decent option (here's a map of the Irish Rail system). It's not a good option for hopping from one coastal place to another (because it's not designed for that). No reason you couldn't take the train out to some point where you pick up a car and motor onward.
There are buses that depart right from the airport in Dublin to destinations in the west/southwest. Cheap, comfy, convenient. Better (and less dangerous) than coming off a trans-Atlantic flight exhausted, jumping in a car and driving off. We took the bus from the airport (Dublin) to Galway, picked up our car there (a few days later), drove all around the west/southwest, returned our car in Cork, and took the train back to Dublin. That worked beautifully.