Advice on itinerary for one-week trip in Ireland

IMHO Public transportation in Eire is not the best.

Dublin is worth 2 nights at most. Avoid Temple Bar unless you like loud noise and drinking.

You’ll need to reserve to see the Book of Kells.

Why the interest in Kilkenny?

Have you considered Belfast? Easy bus or train ride from Dublin The Titanic Museum there is very well done?

Dublin to Galway is about 2.5 hours.
Dublin to Kilkenny is about 1.5 by train. Nearly 2 hours by bus.
Dublin to Cliffs of Moher 5 hours

One way to maximize your time and avoid backtracking would be to head straight to Galway upon landing in Dublin. The bus picks you up right outside the Dublin airport terminal and runs every 30 minutes. You don't need to pre-book, you can either pay on the bus or at the kiosk. The bus drives through Dublin City Center on its way to Galway so you'll also get a little tour of the city. The Cliffs of Moher are a day trip as are the Aran Islands so spending 3 nights in Galway might work best for you.

In case you didn't know, the books have been removed from the Trinity College Long Room (the Old Library) upstairs from the Book of Kells. So that might be a disappointment if you had planned on seeing them.

Not too ambitious, but not too leisurely, either. Ironically, to go between Galway and Kilkenny by train, you have to connect in Dublin, so it’d be an L- shaped trip, not a triangle. Maybe fit Dublin in the middle of your trip? Or just at the end - skip the first night.

Are your flights set? The Shannon airport is closer to Galway, so flying into Shannon and out of Dublin (or vice-versa) could be more efficient, if the price and other factors don’t make that an unreasonable option. Our 2019 trip, we did fly in and out of Dublin, but when we landed, we immediately hopped onto a bus for Galway, and spent our first night there. There are two bus (“coach”) companies: Aircoach and Citylink. Either one gets you to directly from Dublin airport to Galway. Otherwise, use the Shannon airport at the beginning (or end) of your trip to fly into (or out of).

For Aran Islands, stick with just Inishmore, the biggest, given your timeframe. Rather than the ferry, which is more dependent on calm seas and can be interrupted (happened on both of our trips), opt for the much faster, more reliable, more convenient, and only a bit more expensive flight on Aer Arran airline. The airport is west of Galway, but much closer than the ferry terminal. I presume you’re doing this as a daytrip, and returning to Galway for the night.

I’d consider giving Galway an extra night, and Dublin one less. With 2 nights in Kilkenny, that would make 3 each in Galway and Dublin. If you were to reduce those nights even further, you could squeeze in Cork, south of Kilkenny, which has its own airport. Maybe fly into Shannon, stay in Galway, then stay in Dublin, then Kilkenny, finish in Cork and fly home???

Since this trip is a month away, I expect the OP's flights are already set. And, with great respect to Claudia, I think Dublin is worth more than two nights -- ideally you could spend a week there with some day trips. And I wouldn't recommend trying to "fit another city" into this short trip.

I agree with the idea of taking the bus straight from DUB to Galway on the day you arrive, rather than spending the first night in Dublin. Here is the Aircoach schedule, and here is the Irish Rail schedule. Aircoach doesn't appear to serve Kilkenny, but you can get there by train from Galway (changing at Kildare) in 4+ hours, or from Dublin in 1.5 hours. Unless Kilkenny is especially important to you, you might have a better trip basing just in Dublin and Galway -- or visit Kilkenny as a long day trip from Dublin.

It is an intensive itinerary as two nights only effectively gives you a day and a bit in a place. Don’t even think about trying to fit in another day!

You can do an Aran Islands and Cliffs of Moher cruise from Galway

It depends on the journey whether you change at Kildare or Dublin for the Kilkenny to Galway train trip.

Some trips you have to change additionally at Athlone or Portarlington.

What you could do on arrival day is to go straight to Kilkenny- any of many buses to Dublin then the Dublin Coach/Big Green Bus M9 Cork service to Kilkenny-https://www.dublincoach.ie/all-timetables/m9-express-service

or bus from Dublin Airport direct to Heuston, then the train

or, even better JJ Kavanagh #717 direct from Dublin Airport to Kilkenny- every 2 hours- https://bustimes.org/services/717-dublin-airport-clonmel-ms-davis-road

Then on 9/25 to Galway by train-
0728 arrive 1152 via Heuston, or 0830 arrive 1344 via Heuston (so the early train is better)-

then 2 nights in Galway then train or bus to Dublin.