With two full weeks, I'd say spend some time in Dublin, some time in Kilkenny, some time in Dingle, and end in Galway. Fly into Dublin and home through Shannon, if possible. Don't pick up your rental car until you leave Dublin, you won't want or need a car in the city itself.
On your way from Dublin to Kilkenny, make stops at the Powerscourt house and gardens, as well as at Glendalough - at the latter, see the ruins and do some hiking. This makes for a very full day, so get a good start.
On your way from Kilkenny to Dingle, stop at the Rock of Cashel.
On your way from Dingle to Galway, stop at the Cliffs of Moher and drive through/make stops in the Burren.
In Dublin, don't miss Kilmainham Gaol and the archaeology museum (the latter is free). From Dublin, do a day trip to the Bru na Boinne. In Dingle, do the Slea Head Drive and consider a day trip to Skellig Michael. In Galway, consider a trip to the Aran Islands for a day, and/or into the heart of Ireland to Tullamore (castle and whiskey tasting center) and stop at Clonmacnoise (ancient monastic ruins).
Honestly there is so, so much to do. These are just suggestions, by no means an all inclusive list.