Is it possible to take a bus from Shannon Airport to Dingle AND is it possible to rent a car in Dingle or Tralee?
Bus Eireann has a service counter at Shannon Airport, but if no one is there you can purchase a ticket from their vending machine. You will change buses in Limerick and Tralee. Here is a link to the Bus Eireann network map (http://www.buseireann.ie/pdf/1225973201-2008-English-NetworkMap.pdf)
all schedules are in PDF on their website.
I don't know about car rentals, we used only public transportation in Ireland. There are several options to tour Slea Head, etc without a rental car. Enjoy your journey!
For bus timetables and fares for Shannon Airport to Dingle, use the Journey Planner at www.buseireann.ie.
If you google "car hire Tralee" (without the quotation marks), you'll find some choices. However, you'll have a lot more choices if you pick up your car at the airport.
Check www.autoeurope.com, a consolidator that finds you the best rate from among the companies with which it has contracts. If you find a better rate on your own, AE promises to either match or beat it.