I have only been to Dublin but there are some things there I would recommend.
1) Go by Temple Bar, don't bother spending too much time there, it is an obvious tourist trap but it's nice to have been there at least.
2) Go to the Guinness Museum, it is... unique. The only museum I have ever been to where they gave us beer. Twice.
3) Take the train out to Howth (just half an hour or so and the trains are cheap) and walk along the seashore, it is beautiful and there are a lot of really good restaurants that cook up fresh fish.
4) My favourite place to drink and listen to music to at night was O'Donoghues. (The place the Dubliners first became famous).
5) Visit the Trinity College and look at the Book of Kells, if you like old style over the top fancy books it is probably the fanciest I have seen. The old library of the Trinity College is quite impressive as well.
6) If you are interested in some history, you could go to something like the Glasnevin Museum. I also recommend taking a stroll to the Garden of Remembrance.
7) Catch a dinner show at the Arlington Hotel* (they have shows every night and in my opinion the week-day shows are better than the week-end ones). You get some good irish music and at least when I was there, some River Dancing from Celtic Rhytm.
*The one by the Liffey.
That is all I can think of for now.