Overwhelmed - but first decision of when to go and home bases

For avoiding high season consider late May to mid June. Fly into Dublin and give it 2-3 nights, with day trips to Newgrange and Belfast (you can easily do Belfast by train) for example. You could drive to Newgrange and other great sites or simply do a day tour (such as Paddywagon), avoiding the rental car until you depart Dublin-you will not want a car in Dublin.

Head South of Dublin, then West to the Dingle area, finally North to the Cliffs, Galway, and Mayo. Note that Mayo is a fair bit North. Fly home out of Shannon.

http://www.paddywagontours.com/

Scheduling your trip for as early in May as you can manage will help you avoid most of crowds that don't start to arrive until the US schools start their summer vacations in June.
Good advice from Dave regarding booking your trip as an open jaw - will save a day by not having to backtrack to Dublin.
Starting your trip in Dublin (without a car) will allow you to recover from jetlag before you head out on the roads.
Ever driven on the left? If not would suggest booking a vehicle with automatic transmission. The Irish default is manual, ie with a left-hand shifter - an extra fun factor you probably don't want to mess with.
If you elect to fly out of Shannon on your return you might consider heading to Mayo directly from Dublin and then working your way south from there - Killarney isn't far from Shannon.

ummm....now I have complaints from the others about the weather in May.... so it looks like July/August. I like the idea of spending the first couple of days in Dublin & didn't realize we could take so many day trips from there. Would that be more practical than relocating to a hotel closer to the individual places? I will have to get a handle on the distances in Ireland. We have a stick shift here at home & don't think we would have much trouble with that though an automatic would be of course easier in unfamiliar territory. I haven't found an airline from Baltimore or Boston (closest airports with only 1 connecting flight) that flies out of Shannon.

If I 'divided' the country into quarters or thirds to explore each section what places would you suggest?

At the time of year you will be going (peak season, so more options), here are some options for your flights:

Aer Lingus goes from Boston to both Shannon and Dublin.

Both Aer Lingus and Delta go from JFK to both Shannon and Dublin.

American goes from Philadelphia to both Shannon and Dublin.

United goes from Newark to both Shannon and Dublin.

And, American goes from Raleigh/Durham to London Heathrow, where you can connect to a British Air flight (on the same ticket) to both Shannon and Dublin. This is probably fastest, and has the advantage that if your first flight leg to London is delayed, there will be another connection to Ireland very soon. If your flight from Raleigh/Durham to, say, Philadelphia, is delayed, you may have to wait a whole day to connect to Ireland.

So, with one change, you can easily fly Raleigh/Durham to Dublin, then fly back Shannon to Raleigh/Durham, on several different carriers. Just make sure it's all on one ticket, so that it's the airline's responsibility to get you to your final destination. If you're on separate tickets, it's your headache (both logistical and financial) if there's any disruption.

To find prices, times, etc, look at Kayak, then book directly with the airline: http://www.kayak.com/flights

I think your first instinct - to travel during the shoulder season in May - is still sound. The historical weather patterns indicate that the amount of rain will be virtually the same in May versus July or August, although it will be a few degrees warmer later in the summer. May is one of the sunniest months historically as well.
Shoulder season travel will likely save you a lot of money on everything from airfares to accommodation. And of course not having to deal with the crowds in mid-summer will make for a much more enjoyable experience for everyone.
There's a pretty steep learning curve involved in driving on the left the first time, but if you're already familiar with manual transmissions you should do OK. Would recommend starting your driving adventure in the countryside rather than in the city, which can be a white knuckle adventure for even very experienced drivers.