First off, wecome to the forum!

It looks like you will have nine nights. You will be renting a car?

All of your "must sees" seem to be West & Southwest centered.

I would recommend:

June 1 - Arrive Dublin -Collect rental car and headed to Kilkenny via Glendalough and Wicklow (Overnight here)

June 2-4 - Killarney from Kilkenny Via Cashel and Cahir with stops at Rock of Cashel and Cahir Castle (two nights)

June 3 - Gap of Dunloe tour (This is a full day tour)

June 4-7 - Dingle (3 Nights)

June 7-9 - Doolin (two nights) utilize the Kilmer/Talbert Ferry to shorten your drive, take the coast drive up into Doolin with a detour to tour the Loophead peninsula if you wish

June 8 - Take the Cliffs Cruise/visit Inis Oirr

June 9 - Head to Dublin (Overnight)

June 10 - Head for Home

 This itinerary would give you most of what you want to see. If you take the long route from Dingle to Doolin, you could make a stop at Bunratty but I am not sure it is worth going out of the way for on this trip.

Three nights would be a maximum in Dingle becauase of the ground you need to cover.

As you have five people in your party, you will need to make your lodging reservations as soon as possible.

This should get you off to a good start.

Slan Beo (Take Care),

Bit