14 day Itinerary - Chicago to Shannon

My first thought is that there is an hourly bus #51 from Shannon Airport to Cork City bus station, where you have 30 minutes break to catch the hourly bus #226 to Kinsale. Total journey time 4 hours.
Yes a private driver would take 2 hours, so it depends how much time is worth to you.
Probably too late now but did you look at flying into Cork, by connection bus say Amsterdam?

I'm not sure I would go all the way to Westport for one night. Will you still have the driver?
How will you travel between Kinsale and Cobh without the driver? You might want to just hire a driver for the whole trip, Shannon to Shannon. This is not within my financial means, so I have no way of knowing how this is done.

I'd agree with the comment about Westport for one night. It can easily be done as a day trip from Galway.

It might simplify your logistics a bit, and maybe save a little money too, if you opted to fly into Dublin and out of Shannon - it's what's known as an Open Jaw (or multi-destination) flight plan. There's an Express Bus from Dublin airport down to Cork and from there you could connect via taxi or another bus down to Kinsale. Would probably cost a lot less than 250 Euros.
You could then arrange your private drive thru Dingle and the ROK, and thence further up the coast for your night at Ashford Castle.

Hello fellow Chicagoans.

I see the "ouch" in your comments. In my five visits to Ireland, I've used Irish bus (https://www.buseireann.ie/) and the Irish train service heavily, as well as renting a car twice. If you want to save some money, the bus service is efficient. And it's a fun bit of authentic Ireland. (Since you'll only be in the West, the Irish train service probably wouldn't help you, unless you head to the East from the West.)

For Dingle, one suggestion is to walk a section of the Dingle Way (https://www.dingleway.com/). I walked the entire 100 miles of it in 2010, and it was fabulous. Another idea is to cut out either the Ring of Kerry or Dingle Peninsula if you want more time for other things. Though Kerry and Dingle are stunning, perhaps you could use the time for something else.

If you want to venture a bit farther north of Galway, another idea is to visit the Ceile Fields in north County Mayo: https://ceidefields.com/. The Ceile Fields have more stunning views than the Cliffs of Moher, though those are also nice, and there is some amazing ancient history there. If this appeals to you, you could cut the Cliffs of Moher.

A private driver sounds interesting. If you are taking buses I would strongly suggest making sure it is not the Big green bus. We had three very terrible experiences with them in Aug. we found public transport not as easy as we thought it would be in Ireland.

Enjoy

You're aware, I trust, that Cobh, Kinsale and Cork are on the east side of Ireland. We had a car, and even staying in Tipperary (not something I'd do again), it was a bit of a haul to get to Kinsale. With that said, Kinsale and Cobh were, to me, the most beautiful places I saw in Ireland. We didn't do the Ring of Kerry or the Dingle Peninsula, though; we'll need to visit those when we return.

By the way, when you're in Kinsale I highly recommend taking the time to go further out, to the Old Head. If you don't have tee times for the golf course you won't be able to actually enter the massive diamond-shaped property, but the views from its entrance are stunning.

Thank you everyone for your suggestions! We are all set for our trip thanks to everyones helpful advice.