Public transport is not a strong point in the West. Trains arrive from Dublin on separate lines to Ennis, Galway, and Ballina via Westport, but do not run between these towns.
The bus network is more flexible, but there are not many services a day, and you will need to do some serious planning to get around.
Both the Burren and Connemara can be explored on guided day trips from Galway City. But to do full justice to the region, you really need a car—and a good map.