Shannon is about 2 hours away and has a range of US flights in the summer.
Amazingly for a major tourist destination, Killarney has no bus connection to any of Cork Airport, Shannon Airport or Dublin Airport, so if you aren’t hiring a car, it’s an awkward journey of changing buses and/or trains. In that case the connecting flight to Kerry Airport is definitely worth considering.
You have to change at Red Cow Luas on that service.
You can fly from multiple US airports to Dublin, there are even direct flights. If you want to see both Dublin and Killarney you can then fly from Dublin to Kerry Airport, there are 1-2 flights per day of under an hour duration. It will not be a problem to sort transport from Kerry Airport to Killarney which is approx 15 mins away, your hotel could arrange this for you if necessary. Other option as mentioned is to fly into Shannon which is under 2 hours from Killarney, however if you wish to visit Dublin then flying into or out of there is probably your best bet. Renting a car in Ireland is perhaps the easiest way to get around.
PS If you have time I would also highly recommend visiting Dingle which is approx 1 hour drive from Killarney!