We will be flying from Heathrow to Dublin and spending a few days. Then we plan to head to the Cliffs of Moher and the Dingle Peninsula for three more days. Is there any reason we should go back to Dublin to fly back to Heathrow? Or could we fly from Shannon to Heathrow and save ourselves the additional driving?
I would fly into whichever was cheaper, and out of the other one to save backtracking drive time. Either way, you won't want to have a rental car in Dublin, so if you fly into Shannon and out of Dublin, drop your car before spending your time in Dublin. If you fly into Dublin, do your sightseeing first, then rent a car when you are ready to leave town.
In May we flew into Dublin and home from Shannon. It was convenient to build an itinerary that didn't loop all the way back to Dublin. There was a drop fee for the rental car of 70 Euros for the one-way rental.
You can fly from London Luton or London Stansted to Kerry Airport.
Kerry Airport is 20 minutes from Killarney or 1 hour from Dingle
In June we flew into Dublin, spent two nights, then took the train to Limerick and the #51 bus to Shannon Airport to pick up rental car. Returned car at Shannon and flew on from there. This worked out very well and the roads around Shannon were not crowded.
We flew into Dublin and out of Shannon and it was easy and convenient.