Which towns in the south and west of Ireland

Martha, a little more information would be helpful:
How much time do you have?
First trip or a repeat visitor?
Will you have a car and, if so, have you driven on the left before?
What are your interests?
How many people in your group?

Hello, Robert,
Three of us in the group have been to Ireland many, many, many, years ago. We have perhaps one to two nights in each location. We will have a car and have driven on the left before. We like music, history, and value visual loveliness. :)
Thanks for your help,
Martha

OK, here goes:
Unless you're extremely confident that you'll arrive in Dublin fit to drive after the long overnight flight most here would advise waiting a day before getting behind the wheel. One suggestion would be to consider catching the direct express bus from Dublin airport to Galway, spending the night there to shed some of your jet lag, and then collecting your car the next morning after a good night's sleep in a real bed. The one-way drop charge isn't exorbitant and might make your overall logistics a little easier.
Regarding places to stop along the way - here are a few ideas:
Doolin is a favorite here for the trad music, as well its proximity to the Cliffs of Moher. A great way to see them (and avoid the crowds) is to book a late afternoon boat trip - to just view the cliffs themselves or as part of a day trip over to the Aran islands. The boats that depart from Doolin are small however and the crossing might be affected (ie cancelled) due to bad weather or high sea states.
The drive south from Doolin along the coast is one of Irelands great scenic drives on a sunny day. You can either cross the Shannon on the Tarbert ferry or swing over to Limerick if you're a fan of the novels of Frank McCourt. There's a cute little museum there located in the old school that he talks about in Angela's Ashes.
The Ring of Kerry and Dingle are particular highlights, and shouldn't be too crowded yet in May. I'd avoid the tourist trap of Killarney and suggest instead finding a place along the stretch between Glenveigh and Castlemaine - basically midway between both the ROK and Slea Head drive, making it a simple matter to explore both from a single base. From there you can also drive the Skellig Ring Road near the village of Portmagee - another very pretty scenic excursion, with a stop at the Kerry Cliffs and its picture postcard views of the Skelligs being the highlight.
Kinsale is another favorite here, as is the village of Cobh (formerly Queenstown) which was the Titanic's last port of call before her date with the iceberg.
We liked the market town of New Ross. There's a replica of a famine ship there that's worth a visit, and the JFK Arboretum and family homestead is nearby. It's just a very pretty area.
Further along towards Dublin is the monastic site at Glendalough in the Wicklow Mountains. If you can manage to arrive early (before 9 AM or so) you'll have it all to yourselves since the tour buses don't start rolling in until 9:30 when the visitor center (and bathrooms) open.
If you're into scenic, off-the-beaten-path excursions consider picking up a copy of "Backroads Ireland" by DK publishing. Can find used copies on Amazon for just a few dollars - it's the only pub we bring with us aside from the RS guide.

Waterford, Cobh, Kinsale, Baltimore, Bantry, Dingle, Dunquin, Ennis, Doolin, Kilronan (on Inishmore) and Galway would be on my list for consideration. All are in scenic coastal areas, and Dingle, Doolin and Galway are some of the best places for trad.

If you don’t have it already, get a copy of the RS Ireland guide. You can probably get a copy at your library, if you don’t want to buy it. Encourage your adult children to take a look at it, so they can participate in the planning. I did a similar trip a few years ago with my three adult daughters. Two of us were interested in the planning and the other two were happy to go along for the ride. It was a great trip! There are lots of suggestions for routes and where to stay in the guide.

I can't thank you all enough for your help. I so appreciate it. My best, Martha

So, is starting in the northwestern part of our trip and driving south versus starting in the south and driving northward, the best direction? So Dublin to Clifton and south along the coast. Or reverse and come up northward, scenery-wise? Or am I over thinking The whole thing... If don't imagine being able to do this again as a family...
My thanks,
Martha

If your intention is to get your car immediately upon arrival at Dublin airport then it really doesn't matter.
If you want to hold off a day before getting behind the wheel then your best bet is to head west to Galway since picking up a car there is pretty straightforward. There's no easy way to do that if you head south first.
Another advantage of starting in Galway is that you'd have a chance to explore some of Connemarra, which is our favorite part of Ireland. The Loop Drive from Galway, clockwise to Clifden and the Sky Road, further along to the National Park, around to Kylemore Abbey and finishing in the village of Cong is one of our particular favorites.

Doolin is good for two nights if you're wanting to see the Cliffs of Moher and enjoy several pubs with trad music. Add a third night if you want to take the ferry out to the Aran Islands. The drive north along the coast at the edge of The Burren is beautiful.

Dublin > Glendalough > Kilkenny > Rock of Cashel > Kinsale > Kenmare > Ring of Kerry > Dingle and peninsula > Cliffs of Moher.

I can't thank you all enough for your help. I really appreciate it.