Looking for thoughts on this plan.
Constraints:
- Travel party of 4 (2 adults + 3 yr old girl + 6 yr old boy -- kids are fairly seasoned travelers....but they are kids)
- No plans to rent a car (given the kids) so all public transportation
- approx. 12 days in early August (fares look good for roughly Aug. 4-16).

Day 0: Leave US
Day 1: Arrive Dublin, Train/Bus Dublin-->Belfast (Belfast is on the "must have" list by the wife...)
Day 2: Belfast
Day 3: Belfast (Giant's Causeway?)
Day 4: Belfast-->Dublin Train
Day 5: Dublin
Day 6: Dublin
Day 7: Dublin --> Galway Train
Day 8: Galway-->Doolin (bus)
Days 9 and 10: Cliffs of Moher and Aran Islands
Day 11: Doolin-->Shannon
Day 12: Fly home from Shannon

McCombs does great day tours from Belfast, they even have a small bus Game of Thrones tour. We took the bus there directly after landing in Dublin. There are two buses. We took the first one leaving. There is a Premier Inn close by the bus/train station. I wanted Europa hotel but $$$. I like Galway tours for day trips. There is a ferry close to Galway, Rosseveal that can get to Inishmore. The ferry has an office in Galway. They provide transportation to ferry. I love this island best of all the ones that we have visited. Great fort and cliffs are better than Moher.

I love Doolin. You can even horseback ride there. Lahinch is a great beach town for the kids.

Slight itinerary update:

Day 0: Leave US
Day 1: Arrive Belfast
Day 2: Belfast
Day 3: Belfast (Giant's Causeway?)
Day 4: Belfast-->Dublin Train
Day 5: Dublin
Day 6: Dublin
Day 7: Dublin --> Galway Train
Day 8: Galway-->Doolin (bus) OR stay in Galway (Aran Islands)
Days 9 and 10: Cliffs of Moher and Aran Islands
Day 11: Doolin-->Cork
Day 12: Cork
Day 13: Fly home from Cork

Only day 1 (arrive in Belfast) and day 13 (leave from Cork) are set in stone. So others may still change.
Now working on hotels...

Updated plan -- feedback?

Day 0: Leave
Day 1: Arrive Belfast (staying at Maldron Hotel Belfast)
Day 2: Belfast sightseeing
Day 3: Giant's Causeway (question: take an organized tour or just take the bus and tour by ourselves?)
Day 4: train from Belfast to Dublin (staying at Maldron Hotel Pearse Street)
Day 5: Dublin sightseeing
Day 6: Dublin Sightseeting
Day 7: Train from Dublin to Galway (staying at <!--td {border: 1px solid #ccc;}br {mso-data-placementame-cell;}-->Ardawn House)
Day 8:: Aran Islands visit? (any feedback/suggestion on what to focus on)
Day 9: Bus from Galway to Doolin (staying at Half Door B&B)
Day 10: Cliffs of Moher
Day 11: Doolin-->Cork travel (staying at ???)
Day 12: Cork sightseeing
Day 13: Fly home

you can use Public Transport to get to the Giants causeway (which is 1hr from Belfast) I am from Northern Ireland and live ten miles from the Giants causeway
what you do is this ...1. Get the train from either Great Victoria Street or Lanyon Place train stations in Belfast get the train going to Londonderry/Derry or Portrush and get off in COLERAINE (My home town)

2. go into the adjoining Bus station (which is in the same building as the train station turn left once off train platforms and walk a few yards past the coffee shop this is you in the bus station you then get the Ulsterbus Causeway coast rambler to Ballycastle (Service number 402 ) consult the electronic departure boards for the next departure this will drop you off yards from the visitor centre in the Giants causeway, buy a zone 4 I-link card (£16 for 1 days unlimited travel anywhere in Northern Ireland~ remember Northern Ireland is a part of the UK so you will need UK pounds not euro) this card gives you unlimited travel on buses and trains anywhere in NI for one day this saves you £30-£45 as the bus tours from Belfast charge more) go to www.translink.co.uk and plan your journey you will need the Belfast to Derry train timetable and the derry to belfast train timetable for coming back (click rail timetables they are the Number 3 ones) .