Cities

Use a city as a bed and a station, then leave it. The island is small enough that one base can cover a few days of roads.

Coloured houses and the cathedral in Cobh, County Cork

Dublin

Most long-haul flights and many ferries land you here. The city is a rail and bus hub for the rest of the island. If you have two nights, walk the river and one inner neighbourhood. Then get on a train. Dublin can eat a week if you let the museums stack up. That is a different trip from a coast trip.

Galway: West-coast base and the street where this desk sits. Trains from Dublin Heuston. Buses onward to Connemara. The Latin Quarter is short. The useful part is the bus station and the road west.
Belfast: A compact centre with rail to Dublin and Derry / Londonderry. The Causeway Coastal Route is the usual next move. Currency here is the pound sterling.
Cork: A southern rail end from Dublin Heuston. Use it for West Cork and the first piece of the Wild Atlantic Way if you flew into Dublin and do not want to backtrack through Galway.
Derry / Londonderry: Walled centre on the River Foyle. The dual name is in ordinary use. From here the north coast runs east and Donegal sits a short drive west, across the border.