Good recommends for Dublin. Stay 3 nights. On one day train or bus 2 hours
( each way ) to Belfast. See the Titanic Musuem, take the Black Cab Tour of The Troubles and visit St Georges market then back to Dublin.
Next morning Rent the car at the airport.
Things to see outside of Dublin:
Newgrange
Glendalough
Rock of Cashel
Personally I'd by pass Cork and drive to Dingle. 2-3 overnights there. Visit the Gap of Dunloe, Killareny National Park and the Muckross House, drive Slea Head ( head out early so not trapped behind the tour busses). From Dingle get another early start and drive to Burren National park and then onto Gallway. Overnight then back to Dublin.