The best route depends on what you want to see: landscapes, Irish music & dance, pubs, food or Ireland's past.
Day 1: A full day in Dublin.
Day 2: Take M7/M9 to Kilkenny, see Kilkenny Castle. Continue on M9 to Waterford, continue on N25 towards Cork City.
Stop before Cork to see Cobh, last departure of Titanic. Cork city centre can be a bottleneck, take N40 to N71, detour to Kinsale for an overnight stay.
Day 3: N71 to Rosscarbery to see Drombeg Stonecircle.
Detour to Mizen Head, but beware of time, roads get slower the more west you go. Follow the N71 towards Bantry.
At Glengariff: decision between Ring of Kerry (N70) or Ring of Beara (R572, R575, R571). Both are safe bets, but Beara is less commercialized though time consuming if you go all the way to Dursey Island. Stop in Kenmare.
Day 4: Follow the N71 through Killarney (can be a bottleneck). If you do the Ring of Kerry, you can still finish in Killarney, otherwise follow the N22 to Tralee. Continue on N69 to Tarbert and take the ferry to cross the Shannon Estuary.
Follow the N67 to the Burren. Make a stop at the Poulnabrone Dolmen and (time permitting) a detour to see the Cliffs of Moher.
Follow the N67 to Galway City. Visit the city centre for a cultural spree and stay near the city.
Day 5: You don't mention Connemara, Mayo nor Sligo. Connemara is worth seeing though time consuming (a day).
From Galway city follow the N17 north to Sligo (alternatively the N84 to Castlebar, then N5, N58, N26, N59 to Sligo). Sligo offers a great numbers of things to see but it depends on taste. You could stay near Benbulben just north of Sligo Town.
Day 6: From Sligo follow the N15 and enter County Donegal. Follow the N56 west to visit Slieve League. Follow the N56 northeast to Glenveagh National Park. If you stay in a village near the coast you might get lucky to hear people speaking Irish.
Day 7: Follow the N56 through Letterkenny and then return on the N15 to Sligo. You might want to do an alternative route to drive through County Derry and County Fermanagh in Northern Ireland as long as you find your way to the N4 from Sligo to Boyle. Stay somewhere near Boyle.
Day 8: In Boyle find the N61 to Tulsk and turn left on the N5. Stop in Strokestown to visit the Irish Famine Museum. Head back towards Tulsk (only 5 miles) and find the N61 south to Athlone. Find the M6 east. Make a detour at Kilbreggan to visit Locke's Whiskey Distillery in Westmeath. Head south 10 miles for a detour to the next distillery: Tullamore. Then find the M6 back to Dublin.
Day 9: Use this day to adjust the above routes including detours or use the day to visit Newgrange or Glendalough.