Has anyone on the RS board tried taking the bus upon arrival from Shannon or Dublin directly to Galway upon arrival? (We are planning a trip for the beginning of May.) If so, can you please advise for/against that as a plan? Looks as if you can buy the ticket but not reserve a seat on the Ireland national bus system site: http://www.buseireann.ie/ I took a bus from Cork to Kinsale on the tail end of a business trip in 2010 and found it to be really efficient. But if we land in Shannon and can't get a seat for four hours after arriving at 12:15 p.m., I may be in trouble as the designated travel planner :). We would also consider the train from Dublin airport directly to Galway, but that somehow seems more tenuous.
I always recommend public transportation and Ireland has reliable systems. My wife and I took the train from downtown Dublin to Galway in 2014 and it was incredibly cheap and efficient.
I wouldn't worry about getting a seat as long as you have a bus ticket. They're not going to over-sell the bus.
Not sure about Shannon connections, but it sounds like you have flexibility as to where you land. Shannon, of course, is much closer to Galway.
Having a direct rail link from the Dublin Airport to Galway is a huge advantage. If you buy tickets in advance (and get the discount), you have to take the exact train you booked, so flight delays can be a serious problem.
Have fun.