Travel
FLIGHT
Flight to Dublin airport, Shannon airport, or Knock airport (London origin)
PRIVATE transportation
There is a car hire available in all airport. Please make sure you get a letter from your credit card company showing that they insure your car in Ireland (must be written specifically for Ireland). Most insurance company will do it, you just need to ask for it. It can save you a couple of hundred euro.
PUBLIC transport
Dublin Express operates a bus from Dublin Airport T2 to Dublin Heuston Rail Station every 30 minutes. Tickets cost €9 and the journey takes 34 min.
From there, take a train to Westport. The average train between Heuston and Westport takes 3h 15m and the fastest train takes 3h 11m.
Timetable: https://www.irishrail.ie/IrishRail/media/Timetable-PDF-s/Heuston-timetables/10_dublin-westportballina.pdf
The price is around 17-18 euro. Check here: https://www.irishrail.ie/en-ie/accordion/pdf-timetables/dublin-heuston-westport-and-ballina