Spending two nights in Dublin makes sense - not only does it give time to explore the city, it also allows time to recover from jet lag before picking uip a hire car.
A suggested itinery (truying to minimise the number of different overnight stops......
Days 1 & 2 Dublin
Day 3 pick up hire care and drive to Galway for three nights. Visit the Clonmacnoise monastic site with its round towers, ruined chapels and carved crosses. It is a lovely setting on the banks of the Rive Shannon. Also make a stop in Athenry - a walled medieval town with ruined castle and priory.
Days 4 &5 Explore Connemara - One day - Roundstone, Clifden with the Sky Road Drive, Killary Harbour (go for a cruise), Kylemore Abbey Gardens... Another day take a day trip to the Aran islands
Day 6 Head to either Dingle or Killarney for three nights via the Burren (an area of limestone scenery and completley different to anywhere else in Ireland. Take the route round the coast via Dunguaire Castle, Doolin, Cliffs of Moher and use the Killmer to Tarbert ferry.
Day 7 & 8 Spend a day doing Dingle and the Slea Head drive. Another day either spend time in Killarney National Park (Ross Castle, Killarney Lakes boat trip, Gap of Dunloe, Muckross house and gardens, Muckross Historic Farms) OR do the Ring of Kerry.
Day 9 Head back to Shannon Airport via Trallee, Ardare - both attractive small towns - or visit Bunratty Castle and Folk Park..
Day 10 Fly out of Shannon...