If you are driving to Ireland you don't need a passport but you do need some form of ID. I used my work pass! Not sure about the kids but would suggest they would not require anything.
The coast, particularly West, is preferred by me. From Giants Causeway round to Donegal is all stunning. Connemara is wild and wonderful if you get the weather and Co Clare near The Burren equally so (Try Lahinch perhaps). Then you head towards to Killarney and the Beara peninsular. The Islands, Achill, Arran and Clare all worth a visit.
Kids love the seaside and the mountains are near. People really friendly everywhere. On balance within distance of Galway would be good. You have a city near by and excellent scenery / coast to North and South of the city.
It is pricey though. 5e for Guinness in Dublin!