Many good recs here.....I'll add a few:
Spend some time in Galway....not only is it a great base for exploring the Burren and the Connemara, but the town is basically Ireland's version of Austin (imho). The downtown area has a lot of good pubs and several feature traditional Irish music (trad). Generally, the Western side of the island from Kerry up into Donegal has better trad than say Dublin and Galway is a great point to start. Only downside to Galway (and the West of Ireland in general) can be the weather.....tends to be wetter and cooler than say Dublin and the Southeast of Ireland. If you really want to do the medieval banquet thing, the one at Dunguaire castle just south of Galway is much smaller and has better food than the massive one at Bunratty castle.
Speaking of Bunratty Castle, if you have kids, that is a great place to spend a half day. Mine loved the castle and walking the grounds. Bunratty is right outside Limerick, which can be a good stopping off point between Kerry and Galway (or Cork and Galway).
Cork is cool city to spend a day in. Go to the Franciscan Well Brewery.....they are churning out some good craft beer. Even if you can't get to the brewery, most places in Cork have some Franciscan Well on tap. Also, the distillery where Jameson (and several other brands, like Redbreast & Green Spot) are actually distilled is in Midleton, which is probably 1/2 hour drive from Cork City Centre.
Between Cork & Dublin, there are some good spots to stop. The Rock of Cashel (in Cashel, County Tipperary) is a cool old church fortress and Cashel is a nice little town. If you hit the rock, stop in for a drink or some food at Feehan's pub which is just a block or 2 from the parking for the Rock. Spent some time there watching hurling and rugby with the locals. Kilkenny is also a cool little town further up that has a cool vibe to it.