The list is long my friend.
Temple Bar in Dublin is an excellent place to get a feel for bitters and the night life!
If your more in to the countryside, Armagh is where my family hails from.
If you want something more historical you can see the Keshcorran caves near Carrowkeel.
The National Famime Monument features a coffin ship as they were called, is at the base of Croagh Patrick in Murrisk.
Of course everyone wants to see the Blarney stone. Despite many claims of its location, the primary one recognized is Blarney Castle, just outside of Cork and built in to the battlements. Be careful when you lean backwards to kiss it
If literature is your thing, Oscar Wilde's statue is in Merrion Square in Dublin.
The list goes on and on. Feel free to drop me a line, I'll be happy to point you in any direction you please. I'd offer to put you in contact with my relatives as tour guides but the remaining may be moving here due to the currently beyond pathetic Irish economic situation. Can give you their number any way if you like, on off chance they dont move