Dublin to Shannon Ireland, but one place more--
 

So far, this is the Ireland itinerary--
DAY 1 arrive Shannon 9:30 a.m.
drive to Cashel, sightsee and stay the night. (Joy Rockside House)
DAY 2 Drive to Glendalough and Powerscourt to sightsee, sleep at Clontarf just outside of Dublin.
DAY 3 Flight to Rome--8 nights there
Ireland PArt 2
Arrive 4:30 p.m. Dublin, again stay at Clontarf Castle, for three nights. Sightsee Dublin and Newgrange/Trim area.
NOw we have 4 nights remaining, and need to end up at the Shannon airport--departure 11:30 a.m. include time to turn in rental car...
We definitely want to see Cliffs of Moher , Aran Isles, Burren. Is there enough to do to stay all four nights there, or shall we do two and maybe two somewhere else? Stay near Galway first, then go to Doolin area? Weigh in Ireland lovers, this is our first time there...

In your place (which I'm not, because I live in Ireland) I'd head for Galway for three nights, giving two full days. You can get a visit to Clonmacnoise in on the way to Galway.

One day I would use to take the ferry from Rossaveal to Inismore. The other day I would use to ramble in Connemara.

After Galway, you could head south into Co. Clare to visit the Burren and the cliff of Moher, then perhaps visit Bunratty Folk Park and Craggaunowen. The last night can be spent within convenient distance of Shannon, for which I would suggest Ennis.

Thanks, Padraig. Any recommendation for Galway accomodation? I looked at Ballynacken castle as a possibility...
We have teenagers, so need a definite activity as opposed to a ramble. How long approximately should we allow for Clonmacnoise and the ferry ride you mentioned?
Maybe we should spend a night on the island?

I'm surprised that mine is the only response you got. We have a great many participants here who are enthusiasts for the south-west, and I thought people might try to send you that direction.

Clonmacnoise: I think most people spend between one and two hours there. It entails a small diversion from the Dublin-Galway road. From Clontarf to Galway, including a stop at Clonmacnoise and coffee break -- seven hours.

Lest I provoke misunderstanding: it's a ferry, but not a car ferry. It's a fast boat and not a huge distance. I think the crossing takes about 45 minutes. You'll need more time to get to Rossaveal from Galway.

Teens cooped up in the back of a car: yes, that adds a dimension to your question. I'll have a think...

I would take Padraig's route from Dublin to Galway. Clonmacnoise is a very evocative place. Taking the ferry from Rosseveal rather than Doolin is a good choice--quicker trip and less weather dependent. I have done both and would choose the Rosseveal trip any time. Once on Inishmore you could rent bikes for a trip about the island to use up some teenage energy. Galway has a very young energetic feel to it so the kids will probably enjoy exploring there.

Then follow Padraig's advice for the last day.

Have fun!