Are you driving or using public transport? If the former, then M6 to near Ballinasloe then across to Birr. Or to avoid the tolls head out M6 then take turn to Loughrea and Portumna.
1. https://maps.google.ie/
2. Click Get directions
3. Enter Galway as starting point
4. Enter Birr as destination
5. Click Get directions and follow them