Kilmainham Gaol: Go early or pre-book tickets.
National Museum Archeology: see the bog people (free).
Hop on Hop off bus: there are at least 2, red and green. I suggest getting on first thing to get a layout of the city.
Day tour to Newgrange
Trinity College: I walked through, did not take guided tour, Book of Kells is nice, the library room where it is located is spectacular
Beatty Museum: free and fantastic display of very old manuscripts, located at Dublin Castle
General Post Office: mail some postcards, see the history, nice area for walking
River Liffey: Nice to walk up and down, see the bridges (ha penny, James Joyce)
Parks: St Stephen's Green, Phoenix (zoo is here), Merrion Square