We made a similar trip 10-day trip in April-May 2015. Was the best trip (out of lots) in our lives! We flew to Dublin, stayed 3 days doing touristy things, then rented a car to go south and west. Do you have places to stay? I highly recommend bed & breakfasts, booked on your own instead of through an agency. There are so many B&Bs, and the owners of the ones we stayed at were so wonderful to us. In Dublin we stayed at a historic hotel across the street from Oscar Wilde Park and near Trinity College. Also we booked a 2-night stay at Ballyseede Castle in Tralee. That was so beautiful!
From Dublin's south Hertz office, we lucked onto a very scenic road to Kilkenny (past Wicklow Mts.). Stayed at a B&B in Kilkenny - go to the pub where the witch statue is and watch the trad musicians and dancers. During our stay we explored the area (Rock of Cashel, Cahir Castle, Leap Castle (!!!), Waterford), then traveled to Tralee. We skipped Blarney and Ring of Kerry, but wow - Dingle Peninsula was the bomb. We followed Rick Steves' recommendations for that tour. We had booked a trip to Skellig Michael, but was unable to cross because of weather.
Do me a favor - go to Skellig Michael and let me know how it is? I want to go back and climb those 600 steps to the top!
From Tralee we traveled to Doolin because we wanted to visit Aran Islands. I don't think I would go back there again. Crossing the Atlantic was long and kind of like a roller coaster. From Doolin, Cliffs of Moher were awesome.
From Doolin we drove back to Dublin to stay at an airport hotel (very nice one). On the way we stopped at Birr Castle, which was super interesting. Castle itself isn't open to the public (the family still lives there), but the gardens are absolutely beautiful and the giant early telescope is really something to see.
Tips on driving: the back roads are tiny compared to what we have here in the states, but it's worthwhile to drive them sometimes for the scenery. You get used to them, and sometimes have to pull off to the side to let oncoming vehicles pass. In all the towns are round-abouts every block or so. They're great! We had fun driving everywhere.
If you can, stop and sight-see unplanned. If you see a crumbling abbey, or castle, or caves - go for it! Those things we did made the whole holiday extra special for us. Let us know any other questions!