A first week that does not try to eat the whole coast
People land in Dublin with a hired car and a map of every headland. By day three they are tired of petrol stations. The island looks small on a screen. It is not small when the road is wet and the next village is a single shop.
Give the first night to Dublin or to sleep near the airport if the flight is late. The next morning, pick one direction. West toward Galway, south toward Cork, or north toward Belfast. Do not try all three.
If you come west
The train from Dublin Heuston to Galway leaves the car out of the first transfer. From Galway you can take a bus or a short hire for Connemara. Two nights here, two on the coast, one back, is already a week.
If you go north
The Enterprise train to Belfast is a cleaner first move than the M1 with jet lag. From Belfast the Causeway Coastal Route is a day if you do not stop at every lay-by. Stay north or come back. Do not add Kerry on the same ticket.
One stretch, walked and driven slowly, is the trip. The rest of the signs can wait.