Advice - first timers to Ireland

Given the high cost of rental cars at the moment you'd be well advised to try to minimize the actual time you need one. With that in mind I suggest that you head to Galway upon arrival at Dublin airport. There's an express bus from the airport itself that others here have recommended or alternatively you could head into Dublin to take the train.
Once in Galway you could explore the city and get a good night's sleep to shed some of your jet lag before collecting a car there. You could avoid another day's rental by heading over to the Aran Islands (Inishmore) either by ferry or by flying - neither of which would require a car.
Pick up your rental in Galway on the 11th for the short drive over to Athlone and your fishing expedition.
After that you could meander south towards Dingle and Kerry and spend several nights there as the spirit moves you. My standard suggestion is to find a central location between Dingle and Killarney so you can visit both as a short day trip while only having to unpack once - might save you a bit of the hassle of having to relocate every other day.
While in early May you'll be ahead of the crowds of summer you're still better off trying to avoid Killarney town as a base - can be pretty congested when the tour buses are running.
From there you could proceed counter-clockwise for quick stops in Kinsale, Cashel and Killkenny on your way back to Dublin.
Know that Killkenny is a short train ride from Dublin (1.5 hrs) so you could make that a day trip if you chose to add a day in Dublin for that purpose.
To be honest, I'd save everything on your list after Kerry for a future trip. The most common mistake for first time visitors is to try to see too much in too short a time period and thus run yourselves ragged in the process. I do speak from some experience here since that's what we did on our first visit years ago. Better to slow down and enjoy the subtle charms of the west of Ireland on your first trip. Add an extra day or two in Dublin for day trips (without a car) to Kilkenny, Belfast, the Boyne Valley or Glendalough and the Wicklow Mountains.
One-way drop charges for a rental car are typically not too bad, so you do have the option of picking up a vehicle in Galway and dropping it somewhere else. One suggestion would be to drop it in Killarney - the Kerry airport there has a couple of cheap flights per day back to Dublin, so if you can work one of those into your itinerary it might streamline things a bit ... saving you time and money in the process.

Skip Galway and from Athlone drive to Dingle; from Dingle drive to Cork; from Cork drive to Killarney; from Killarney drive to Kinsale; from Kinsale drive to Waterford; from Waterford drive to Dublin.

It appears you're going to have 4 days between leaving Athlone and arriving back in Dublin. Maybe look harder at the Galway area for those days including the Aran Islands, The Burren, Cliffs of Moher, and towns like Doolin and Ennis. It will save you a lot of driving time compared with trying to hit everything you want to see further southwest. Regarding basing in Kilkenny, spend your first afternoon there and then get up early on day 2 and day trip down to Waterford (Kells Priory is worth a stop in route). Return via Cahir Castle and the Rock of Cashel - or hit Cahir and Cashel on your way to Athlone the morning of May 11. Whatever you do don't try to see too much or you'll end up spending a lot of your vacation time traveling. Enjoy your adventure!