Nine nights in Ireland

I would say this is too much for only 9 nights. I did three weeks by public transit and the one thing you need to remember is that it's going to take a lot longer to get from place to place, wasting valuable touring time. With only three nights in Dublin they will only have two days to see just the sights in Dublin let alone a day trip to Belfast. I would leave Belfast off the list. If they plan to go from Dublin to Killarney by train this will end up taking up half a day. When they get there they will be on foot unless they can find a company that does half day tours that start after lunch. Most of the guided tours are day trips that leave town around 9am. The same can be said for Galway, although it is a shorter train trip from Dublin. A rail pass usually allows a limited number of travel days within a time frame so I'm not sure it would be economical for them. Quite frankly, if it were me, I would pick just two places for such a short time without a car. Either Dublin-Killarney, or Dublin-Galway. They don't want to be spending all of their time on a train getting to the next town only to have limited time to actually see the sights. I spent four days in each of Dublin, Killarney, Galway and Belfast and still found myself rushed. You need to go about planning a trip like this in the reverse order. Instead of deciding on how many nights you are going to spend in a place go first to the train schedules, and web sites for the companies that do the day tours for the places they want to see. Make your travel plans around these schedules. For instance, what is the earliest train from Dublin, what time does it get to Killarney, how long will it take them to check into accommodation, then is there a guided tour they can catch once they are settled? Just as an example I googled Killarney half day trips and came up with this: https://www.viator.com/Killarney-tours/Half-day-Tours/d911-g12-c95?pref=02
Remember, too, that the day they arrive and leave their schedule will be dependant on getting out of the airport into Dublin, or what time they have to be at the airport in Shannon (and how long it will take them to get to Shannon from wherever they are). Do not plan on any guided tours on days of arrival and departure, and try to arrive early in the day, and leave late in the day if at all possible.

If traveling between Killarney and Galway by train, the travelers need to "sign off" on the schedule plan. Northbound, the departure that requires only one connection departs at 7:40 a.m., arriving 13:40 (6 hours). Southbound, all departures on my test date require either two or three connections. www.irishrail.ie

I agree with Anita that it is far too much ground to cover in 9 nights using public transportation.

I would go with Dublin and Galway and do say trips from there.