Fly in to The Dubs and flyover out of Shannon.
1st option: fly in & out of The Dubs Base yourself out of Dublin - 2 days to see the sights. The remaining time take tours via bus South to the Wicklow Mountains (Powers Court) and bus a bit north to Newgrange. You can get tours on O'Connell Street.
2nd option: fly into The Dubs and fly out of Shannon. 1st day Dublin. 2nd day pick up rental car travel to Galway - then time to stroll Galway & enjoy the sights there. 3rd day Cliffs of Moher & environs. 4th Day: tour Connemara. 5th Day fly out.
3rd option: fly into The Dubs and fly out of Shannon: 1st day: stay at an airport hotel that has shuttle to and from airport (Blewelys is great) after check in head back to airport for bus or taxi into Dublin. Hop on Hop off your best option for optimum day in Dublin. 2nd day go to airport pick up rental car and hit the take the M50 and exit onto the N7 Limerick and South of the N21 to Dingle stay 2nights. 3rd day Dingle - you will have a full day in Dingle if you make it in the night before - 4th day head North to the River Shannon and take the Tarbert-Kilmer ferry and onto Cliffs of Moher - that night stay near Bunratty castle which is near Shannon airport - day 5 fly out of Shannon.
4th option: Fly into and out of Shannon: day 1 pick up rental car drive to Galway City. 2day off to the cliffs of Moher via Ballyvaughn - tour the dolman and Burren on your way to the Cliffs. Head south to Tarbert-Kilmer ferry - cross the River Shannon - stay in Tralee. 3rd day onto Dingle - 2 nights - it will give you a full day to explore on day 4. 5th day: early morning to Shannon for flight out.
To be realistic Dublin alone minimum of 2 days - which is 1/2 of the time you have allotted. Even with the newer Motorways - - driving on Irish roads do take time - even with a wonderful map and Navigator.
Good luck on your decision.