travel by public transportation

Here are some ideas
https://www.rome2rio.com/map/Galway/Killarney

No matter how you decide to do it, by train or bus, it will be a long trip. After trying to figure it out myself I finally realized it was best to go by train and make couple of stops along the way to break up the journey. So, I ended up going from Killarney to Kinsale to Kilkenny and then on to Galway. If you are trying to get from Galway to Killarney by bus I think you will end up spending a whole day getting there. It might be easier by train even if you have to go through Dublin.

In case this is of any help, the M7 "express" Dublin Coach goes from Dublin to Ennis, Limerick to Killarney. It is a different company than Bus Eireann so is not on the Bus Eireann schedules. From what I saw when looking into it (I gave up on KIllarney) was that Bus Eireann from Galway requires a change in Limerick to get to Killarney.

I was also looking into traveling in Ireland using public transportation this July for 11 days. I was thinking about flying in and out of Dublin and visiting Galway, Cork, and possibly Northern Ireland.

Public transport between Dublin and Galway is excellent as it public transport to and around Northern Ireland. It gets a little trickier when you try to go to the SW

It is often useful to check several bus companies and the train. There is a citylink bus from Galway to Limerick that takes an hour and 20 minutes and the Dublin Coach M7 from Limerick to Kilarney takes one hour and thirty five minutes. I believe the train and bus Eireann take longer.

I visited Ireland in March 2017 on a self-guided rail tour. On one of the days, I was supposed to take the train from Killarney to Galway, but due to a transit strike, the company that put my tour together ended up arranging a private driver for me. So I didn't use the rail system for that leg of the trip. But I looked up the itinerary just now, and the route I was supposed to take would have gone Killarney to Mallow, Mallow to Limerick Junction, and Limerick Junction to Galway, with a train switch at each stop. The total travel time would have been just over 4.5 hours. I assume it's about the same going the other direction.

Just for comparison, I think the private drive took about 2 to 2.5 hours, but I can't remember exactly.

http://www.translink.co.uk/ may be better for finding buses on this particular route, and for some bus/train combinations.