Killarney (Ring of Kerry, Dingle Peninsula) is probably top of the list for most people, followed by Galway and Connemara (doing the Burren on the way). With Dublin they make three good stops.
Include places like Newgrange as a day trip from Dublin (along with Hill of Tara and Trim Castle). Do The lovely early Christian site of Glendalough and Rock of Cashel on the way to Killarney. Spend a day for Rink of Kerry, another day for Dingle Peninsula and a day around Killarney itself with the Gap of Dunloe, Muckross House, gardens, and traditional farms, Ross Castle... )
Spend a day travelling through the limestone scenery of the Burren to reach Galway ( if time allows try and do a boat trip to see the Cliffs of Moher - much more impressive from the sea than on top which is always busy with tourists). From Galway, consider a day trip to the Aran Islands. Drive along the coast to Roundstone and Clifden. From Clifden, make sure you do the Sky Drive which on a good day must rank as one of the best short drives in the world. Best done clockwise). Do Kylemore Abbey if time allows, or do this next day along with Killary Harbour (the only 'fjord' in Ireland) and go for a boat cruise). If you are interested in the way of life, then you would enjoy the National Museum of Country Life at Castlebar. This has a collection of videos taken in the 1950s which are fascinating. There is also Cong which was used for the John Wayne film "the Quiet Man". It also has a ruined abbey and the Monk's fishing station with bell for the monks to ring when they caught a fish...
On the way back to Dublin, make a brief stop at Athenry, a medieval walled town with runed castle and also the wonderful Clonmacnoise, another early Christian site.
The other alternative is to head north into Northern Ireland visiting Belfast for a couple of days, with the Titanic connection and Uster Folk Museum. Then head north up the coast, making a stop at Glenariff Forest Park and doing the wonderful Waterfalls walk. Then head along the Antrim coast and base yourself here for a couple of days. There is Carrick a rede rope bridge, Dunluce Castle, Giant's Causeway, Bushmills Distillery... There is the Ulster American Folk park near Omagh which is fascinating as it looks at the history of families who migrated from Ulster to America, It has reconstructions of their homes in Ulster and also America. There is a ship which carried them across the Atlantic and they had a tiny space to live,and didn't know where exactly they would land or how long the journey would take. There are then reconstructions of their houses in America. You could easily spend a full day here. Continue to Derry and into Donegal. This is less popular with tourists but scenically very good. I prefer Slieve League Cliffs to Cliffs of Moher. There is Glenveagh National Park and castle.
