With just four days I would suggest you base yourself in Galway. It is a vibrant and bust town with good vibes. You could spend a day exploring the Burren with its limestone scenery, completely different to anywhere else in Ireland, which could include the Cliffs of Moher.
Spend another day exploring Connemara which has some of the best scenery in Ireland. Head to Roundstone with its tiny harbour and views across to the Twelve Bens, then do the Sky Drive in Clifden. On a sunny day, this is one of the best short drives in the world. You could head up to Kilary Harbour (the only fjord in Ireland) and go for a boat trip. Return via Cong which was used to film The Quiet Man starring John Wayne and Maureen O'Hara. There is the ruined Cong Abbey with its fishing station on the river, with bell to ring to let the kitchen know if a fish had been caught...
Alternatively do a day trip to the Arran Islands.
Head back to Dublin on Day three via the medieval town of Athenry with its ruined castle and abbey and the lovely the early Christian site on Clonmacnoise. Book the Storehouse Guinness Factory tour for the morning of Day 4.
This will give a good flavour of Ireland - its scenery, culture and history.