Manda,
Hello. I have visited Ireland five times and love the country. I actually inquired about emigrating there.
Between Cork and Dublin:
Visit Kilkenny, Ireland (http://www.kilkennytourism.ie/). It's a beautiful medieval town. Sites include St. Canice's Cathedral and Kilkenny Castle, but the town itself is a draw. Rent a bicycle. The surrounding countryside is lovely.
The Wicklow Mountains are south of Dublin. It's worth seeing Glendalough (http://www.glendalough.ie/), the 1,000-year-old ancient monastic settlement in Wicklow. Some of the glacial trenches nearby are beautiful. There are dozens of walking trails. Driving through Wicklow is also beautiful. In 1999, I walked a 100-mile-long trail called the Wicklow Way (http://en.wikipedia.org/wiki/Wicklow_Way) that goes through those hills.
Another post mentioned the Rock of Cashel (http://www.cashel.ie/), the castle that St. Patrick himself reportedly visited. I agree that that's also worth seeing.
I would avoid Waterford, the home of the crystal factory. It's a tourist trap.
Galway: The Galway Arts Fest (http://www.giaf.ie/) is supposed to be a top event, but I have never attended it. It's in July and two weeks long! In Galway I liked visiting the relatively recently completed cathedral (a great example of Celtic art inside, though the exterior is uninspiring), Nora Barnacle's house (James Joyce's spouse) and Galway City Museum. Galway is great as a stepping off point to visit the Burren (http://www.burrennationalpark.ie/), a national park, and for authentic Irish music and dance. Regular people in Galway bars dance and play music. It's an authentic cultural experience.
Aran Islands (http://www.aranislands.ie/): I've been to Inish Mor Island, the biggest, two times. It's superb for bicycling and visiting Irish antiquities, such as Dun Aonghasa, an ancient rock fort. A friend of mine raved about Inish Meain, the medium island, but I have never been there.
Sligo: I great idea is to use this town as a base to visit Croagh Patrick (http://www.croagh-patrick.com/), a famous mountain on which St. Patrick reportedly prayed for the soul of Ireland. It's near another town called Westport, where I have also stayed. I have climbed Croagh Patrick, though the path to get to the top is unpaved. It takes about 2.5 hours.
Nearby Sligo is packed with hundreds of Irish antiquities, such as churches, round towers, dolmens and Norman keeps. I'd be happy living my life there.
Hope this helps.