Hey, I was just in Ireland a month or two ago as well for only 8 days in between destinations. Unfortunately I had to rely on buses which I found to be one of the most expensive ways to travel in any country I have visited as well as ridiculously inefficient for the size of Ireland, but that's travel for you. Very good you have a car. I skipped Dublin mostly and only spent a day there so I really didn't get a great feel for it. It seemed pretty compact though so 2 days might be enough to see the 'sights' if you will. I then went across to Killarney and the Ring of Kerry for 3 nights. I had people saying Killarney was so touristy and not to spend much time there, but honestly places are touristy for a reason. Killarney is beautiful. The short walk out of town to Ross? Castle, the surrounding mountains and lakes, good place to cycle, town is so quaint and tiny, also took a daytrip to Ring of Kerry but would be great that you have a car as there are lots of little places to stop off on along the way. Tip - possibly ask around town as to which direction you should drive around the ring, as my guide on the bus I took had said in the summer it is crazy to drive the opposite way around from the flow of traffic because of all the coach tours and tourists on the narrow winding roads. My driver went north first and then counterclockwise.
I also spent two nights in Doolin. Awesome little village with 3 hostels, 3 pubs with great traditional music, and a few shops. You can also hike to the Cliffs of Moher from town along the sea though it's not necessarily an easy hike and it takes a while but AMAZING views!! Galway was my last stop and it was a really nice town too. Couldn't really say what there is to do there and would likely only spend 2 nights.
Other places I would have liked to go would have been Cork, Cashel, and Donegal. Also a town south of Dublin I think called Westport was recommended to me. Anyways, it's a great country and the people were awesome so have a great time and make sure to drink your share of pints in the innumerable pubs.