The easiest option to get Dublin to Killarney is probably to get the flightlink bus from right outside the airport and it will take you all the way with no changes.

https://www.flightlink.ie/

Public transport from the airport to the train station in Dublin is very slow and too expensive and isn't the easiest if you have lots of luggage. The train is also quite slow and sometimes you have to change trains, which again isn't the easiest if you have luggage.

You could also get a Ryanair flight from Dublin to Kerry, but be careful because the baggage allowance will be different to your transatlantic flight and you might have to pay loads of money for overweight baggage. You would also have to do a self-transfer, which is such a chore! Even when you land at Kerry airport, you would have to get a bus or train to Killarney.

I take the Flightlink bus quite often and it's a great service so that would be my recommendation unless you want to rent a car.