Hi Kevin,
How nice that you are coming to Ireland for such an occasion!
To get to Dublin Heuston (the train station) from the airport, you can take a shuttle bus, which takes about 35 minutes. A few companies operate them, such as
Dublin Express.
Alternatively, you can also take the bus from the airport straight to Galway. There is a direct connection with Citylink (bus 760/761). This journey take about 3 hours.
For more information on itineraries and tickets, I'd recommend visiting the
Transport for Ireland website. Here's also more information on
public transport and
trains in Ireland.
Let us know which journey you end up choosing and I hope you enjoy your time in Galway! Do you have any plans besides the wedding?
Cheers,
Antoine