Route from Dublin to Killarney National Park (via Rock of Cashel)

Looks to me like it's pretty straightforward out of Cahir: stay on the M8 until it becomes the N73, to the N72 and on to the R561 into Dingle. Am guessing that the reason Google maps may be switching back and forth on you is because it's accounting for traffic slowdowns in both Killarney and Limerick - both of which are best avoided if you can manage it, which the above routing will certainly do.
Starting out on the Motorway will allow you to achieve a level of comfort with the whole experience of driving on the left before you have to get off on one of the N or R roads. Might want to preview your route by looking at the street view feature on Google maps ... especially the transition points between the major roads. Always helps me - might work for you too.
And if you're not already aware, try to book a vehicle with automatic transmission for your first adventure of driving in Ireland. The default rental is a manual, and you probably don't need the extra challenge of having to shift with your left hand at the same time you're wrestling with everything else.
Safe travels.

Thanks you made that so easy. Will take all your advise and have the automatic. Thanks again

“Not use to driving on the opposite side of the road.”

The best advice I received when driving in Ireland was to remember to look right when entering the roundabouts. Traffic in the roundabouts have the right of way. Signaling when you exit is always appreciated.

Thanks for that advise. Look RIGHT.

Another thought since you may be new to driving on the left and/or driving in Ireland: The rural roads (the R and L roads on the map) can be a bit daunting until you get used to the local rules of the road. They can narrow in places to only a single lane, meaning that you have to be particularly alert for approaching traffic and be prepared to stop and use one of the pullouts along the shoulder to allow the other vehicle to pass safely. It's not as scary as it sounds, although the first couple of times it happens to you I guarantee that your heart rate will spike for a few seconds. Thing is, the locals deal with this situation all the time and are quite patient when dealing with momentarily confused Yanks. Usually by the time my brain had processed what was going on and what I needed to do, the other guy had already pulled over and was waving me thru. Really no big deal. And it helps to know that the Irish drivers are some of the friendliest and most patient that you'll find anywhere.
Here again the street view feature on Google maps can be of help - can view long stretches of the R and L roads in advance to get an idea of what you're getting yourself into.

Thank you for the advise.