Flights are available into Dublin Airport and then a bus or train to Ballina where you will be picked up. There is a route 22 bus directly from Dublin Airport to Ballina. View timetables and purchase tickets through Bus Éireann.
A train leaves from Heuston Station to Ballina on Irish Rail. Make sure you have enough time between flights as the train station is a good half hour across the city.
Flights into Shannon and Knock Airport are also available. You can hire a car or get a bus from the Airport to Galway and then change for a bus to Ballina or get the shuttle bus from Knock Airport to Charlestown and then a bus from Charlestown to Ballina.