Robert’s itinerary would work well and I love Connemara.
Have you thought about visiting Newgrange while in the Dublin area? It dates from around 3200 BC, making it older than Stonehenge and the Egyptian pyramids.
If heading to Galway, I also recommend the early Christian site of Clonmacnoise with its round tower, ruined churches and carved crosses. Arthenry is also worth seeing - a medieval walled town with a ruined castle and priory.
Galway is a vibrant city. Clifden a lot quieter and more old fashioned. Don’t miss tiny Roundstone, with its harbour and superb views over to the Twelve Bens.
The National Museum of Country Life near Castlebar is also well worth adding to the list of things to do. As well as lots of information and artefacts about the way of life in Ireland 50 + years ago, it also has recordings of people talking about their work, traditions and customs. We found these fascinating. Some of the traditions were decidedly spooky.
Do spend a day around the Burren - can be done as a day trip from Galway or alternatively find accommodation around Doolin/ Lisdoonvara area for a couple of nights. This is limestone scenery and completely different to anywhere else in Ireland. The drive around the coast via Kinvarra to Ballyvaughan to Doolin is some of the best scenery. The Cliffs of Moher are on everyones tick list but are always incredibly busy. They are best seen from below on a boat trip.
From the Burren, you can return to Dublin the 'long way' via Rock of Cashel or Cahir and Kilkenny. Spend a night in Kilkenny as there is a lot too do and see in the town
You won’t cover a lot of Ireland but you will see one of the best parts in some depth and really get an understanding of the country.