Arriving in Kerry and eventually driving to Dublin, where is a good home base?

“Flying into Kerry.”

Puzzled. Kerry is a county.

I see there is a Kerry Airport.

Me, I’d opt for Dingle but what are you wanting to do. Hike, shop, party?

Sorry, I thought it was clear that I would be flying into an airport. Yes, Kerry Airport.

We want to see the sights of Ireland. Nature looks beautiful there, but we won't do any hiking. Just curious about suggestions for first time visitors.

The airport in Co. Kerry seems ideally located for getting to the southwest of Ireland right off. https://en.wikipedia.org/wiki/Kerry_Airport

With only 5 nights, you could sleep in Killarney nights 1 & 2, with the intervening day spent circling either the Dingle peninsula or the Ring of Kerry. Then drive north to Cliffs of Moher and sleep in someplace like Lahinch for your 3rd night. Spend nights 4 and 5 in Dublin.

Personally I would choose the Cliffs of Moher over Blarney Castle and the Cork area, but the latter would be your other choice. I wouldn't try to fit in both on a short trip because they are in opposite directions from your arrival airport.

With only five nights and you’re spending two in Dublin assume that you’ll return. Dingle is atmospheric with good shopping and a fun pub culture that makes it a worthy stop. On day two tour the peninsula.
From there drive to the Rock of Cashel (3h 15m) and then to Kilkenny (1h 15m) and spend one night there before continuing on to Dublin (3h 30m).

In early May you'll be ahead of the big crowds of summer, so staying in or near Killarney would probably be your best option for the 3 nights. From a single base you could explore the National Park, the Ring of Kerry, and spend one day visiting the Dingle peninsula and Slea Head Drive. There's plenty to see and do in the immediate area without trying to explore further afield.
You could save yourselves the chore of driving back to Dublin by turning n your car at Kerry Airport and just catching the train.
Edited to add: if you're OK with driving back Dublin a good intermediate stop would be the Rock of Cashel as was mentioned. Would be a good place to stop for lunch and a tour of the old abbey would make for a good leg stretcher for a hour or so ... maybe a little longer if you included the ruins of Hore Abbey located just down the hill from the main monastic site.
Could return the car at Dublin Airport or possibly along South Circular Road on the outskirts of the city if arriving from the southwest - wouldn't have to negotiate city traffic that way.
And the one-way drop charge for rentals in Ireland is pretty minimal.

Drive less, see more. Killarney or Dingle are both excellent bases for a couple of nights. With only three days, you can wind up spending it all in a car if you try to do too much. The Dingle peninsula with the Oratory is well worth the drive. Ring of Kerry is longer but similar, unless you plan a boat trip to Skellig Michael. The Killarney National park is also worthwhile if you like walking. Muckross house is also a great piece of history. Before you go to out of your way to the Blarney stone, read Rick Steves' critique of that whole experience. But Cashel is well worth the stop on the way back to Dublin.