Dublin's public transport, while still inadequate, has seen major improvements in recent years.
Dublin's DART (Dublin Area Rapid Transit) is a suburban train system that goes from Malahide (north Co. Dublin) to the city centre and then south to Greystones in Co. Wicklow.
Arrow suburban train services serve western suburbs (one line to Maynooth in Co. Kildare and one line to Kildare Town).
The LUAS is a modern tram system with two lines both serving the southern suburbs.
DART information: www.iarnrodeireann.ie/dart/home;
Arrow information: www.irishrail.ie;
LUAS information: www.luas.ie
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Dublin Bus has the cheapest links to/from Dublin Airport if you use the regular, standard bus services (16a, 41, 41b, 46x, 56x, 102).
Try to buy day tickets (or weekly/monthly tickets if you're staying longer) which offer much better value for money and flexibility than buying single journey tickets. You might need to supply a photo and ID to buy some weekly/monthly tickets.
Information on public transport options is available from the web or at the Bus & Rail information desk in Dublin Airport:
www.dublinairport.com/at-airport/airport-maps/arrivals.html
Dublin Bus information: www.dublinbus.ie/home/
Dublin Bus airport services:
www.dublinbus.ie/your_journey/airport_services.asp
Bus Eireann operates long-distance and local routes throughout Ireland and serve some destinations near Dublin.
Bus Eireann information: www.buseireann.ie/
There are many other bus and coach companies that operate services in Dublin and throughout Ireland. The best place to find information on these is at www.boards.ie > Soc > Commuting & Transport > Private Bus Timetables:
www.boards.ie/vbulletin/showthread.php?t=274473