There are city and regional buses in Galway which are fairly reliable. There is also a train service which takes you from Dublin to Galway City, stopping at a number of towns along the way.
The best way to see the county is by car. There are car rental agencies in Shannon airport, just make sure you pre book a car seat if you are not bringing your own. Most parts of County Galway are accessible through National Roads marked “N” on roadmaps e.g. “N85”. Smaller country roads are denoted with the letter “R”. Other roads or tracks may be prefixed with the letter “L”- these roads are narrower and usually not marked.