I'm Irish - and I think the best idea is to try and travel about a bit. To be honest, our economic success in the past 15 years has left Dublin an "average" city at best. People aren't as friendly as they used to be; food, drink & shopping is expensive. However, I do think a few days in Dublin would be worthwhile - even if to see The National Gallery, Kilmainham Gaol, Phoenix Park (and the zoo)and to do a "Viking Splash" tour, which takes you around the oldest parts of Dublin, and ends in a little splash!
From Dublin, head toward Cork - where things are a bit cheaper, and often the food is better. Into the countryside is lovely if you have a car. If you do have a car, the Ring of Kerry is a must - everyone who ever did it enjoyed it. It's a long time in a car, but it is beautiful scenery.
At Cork/ Kerry, you're down the south of the island. From here, it's great to head north along the west coast - there's some beautiful scenery and little interesting tourist spots along the way. The West is mainly about Galway and Connemara (where there's quite a bit of stuff to do, again mainly site-seeing, eating and drinking, but there you go!).
Best thing to do is check out the irish tourist site to see what you might enjoy doing most (http://www.ireland.ie/), but certainly my favourite places are Cork, Kerry, Clare, Galway and Donegal.