(Note that all times and fares below are estimates)
The places I would recommend are:
(Comment - if you can afford the time and the cost you should stay overnight in Belfast or Kilkenny)
Kilkenny is a beautiful Norman-Irish medieval city with a wonderful castle open to visitors. It is also is home to the Kilkenny Design Centre which sells products designed in Ireland.
To get there: By train from Dublin - 90 minutes each way. Trains go from Dublin Heuston station and cost is about €24 return.
Belfast is now an exciting cosmopolitan city easily reached from Dublin - and well worth a visit if you have time. You can visit the Titanic exhibition and learn something of the history of the "Troubles" on various tours. Please remember that you will be entering the UK and will need to use the UK pound currency - easily available from ATM's. Also ensure you know what your visa status is for travel to the UK.
To get there: You can take the train from Dublin Connolly Station to Belfast Central taking about 2 hours. Day return is €40 - but you may be entitled to a reduction.
Brú Na Bóinne incl Newgrange (Wikipedia entry here)
Brú Na Bóinne
is a World Heritage site with a number of passage tombs dating back
5000 years. It is also a "solar calendar" and on the winter solstice the
sun shines into the inner chamber.
To get there: You can take the train from Dublin Connolly station to Drogheda and then take a 163 bus from there to Dunore and the Brú Na Bóinne visitor centre. About 60 mins each way. Fare probably about €20 - check that first.
Tours: It might be easier to research online for tours from Dublin that probably include other sights.