What everyone else said, AND:
1) Oscar Wilde's house. Merrion Square-- not only has some beautiful houses around there, the Georgian area, but also, hey it's Oscar Wilde's childhood home! You can take tours on Monday, Wednesday and Thursday (I think) and it's very beautiful inside. All furnished and restored, original floors, a beautiful stained glass window of the Happy Prince, etc (and I'm babbling.)
across the street is Oscar's statue. And not far away is Westland Row, where he was born. A plaque will help you find it.
2) History museum-- tara brooch, viking ship, gold celtic artifacts, other famous Irish historical material objects. Very interesting stuff. They also have some Egyptian artifacts about which I was pleasantly surprised.
3) I also liked the National Gallery. Close to Oscar's house. I seme to remember a portrait of Bono is in there, done by some guy, can't remember the name, lol.
Also, a lot of the touristy things are on the south side, but the north side is fun, too. Shopping/ pedestrian streets off OConnoll street are good as long as it's sunny.