So if you intend to travel somewhere other than Dublin from Dublin Airport, the handiest and cheapest way of doing so is to catch a coach. This may come as a surprise to you, because I find that the coach services from Dublin airport are not well advertised.
There are two national coach companies in Ireland, Bus Nestor and Citylink. Citylink is the newer and more frequent of the two. Nestor Bus’ coaches tend to be older and less branded (Citylink’s busses are an unmissable bright yellow and blue), but in my experience they arrive sooner and the drivers are often more polite. Unfortunately, both busses run only the Dublin/Galway route (though they both have several stops on the way). Citylink charges €19 from the Airport to Galway and Nestor Bus charges €18.
To find the busses, leave the terminal and continue through the second building. The busses are parked next to each other to your right as you emerge onto the car park.
Citylink also offers a separate service between Galway and Shannon airport (Nestorbus does not). This much shorter trip is an exorbitant €16 one way and €23 round trip.