Train would be a pleasant way to travel across the island. You'll have to consider several transfers to make it happen though. Not tough ones, but unfortunately it's not a straight shot. I haven't gone that route in a while, but here's how I remember it...

You'll have a 45 or so minute ride from Dublin Airport to Heuston Station.

Train to Limerick (I've had to transfer at Ballybrophy before, but it depends on the route/schedule).

In Limerick you'll have to decide if you want to bus/taxi it to Shannon or train/bus/taxi it Ennis (the trainline between Limerick and Ennis/Galway just reopened this year and the schedule is limited - I suppose that new line also means you could go Dublin-Galway-Limerick as well).

You could also consider the bus... it'll be cheaper, you'll see just as much and the buses are pretty nice, and depending on which busline you choose, the schedule can be set to take you from airport to airport without transfers.

I would get the Bus direct from Dublin Airport to Galway and then either train or Bus to Ennis.  JJ Kavanagh run regular buses between Dublin and Shannon Airports.

http://www.jjkavanagh.ie/
http://www.citylink.ie/
http://www.buseireann.ie/

-- Edited by tony2phones on Thursday 17th of June 2010 01:19:42 PM