Travelling Ireland

You need a car for Ireland so I suggest taking a Rick Steves tour: https://www.ricksteves.com/tours/ireland.

Irish Rail and Bus Eireann (both long distance coaches and local bus routes) together have an extensive network, which reaches many places.

It would be useful for you to provide a desired itinerary and then it can be properly assessed to see if if it is workable, or not, by bus and train.

Here are their websites-

https://www.irishrail.ie/en-ie/
https://buseireann.ie/

Tours don’t suit everyone, and learning to drive on the left doesn’t suit everyone. I did three weeks in Ireland with a bus/rail pass and loved it. You need to plan carefully to make the most of your time, though, and it is best to base yourself in larger towns that will have a good selection of guided day trips to get you to sights you won’t be able to get to by public transit. Start by looking at the available day trips from Dublin, Galway, and Killarney and then figure out how many days you will need in each place to see the sights that interest you. Then you can add some stops in smaller towns in between those places. There have been several people ask this question here so I would use the search function here to find those posts as there were some good answers from some well travelled people.

Depending on how much time you have and your particular interests you can craft a very nice first-time itinerary by making selective use of public transportation - basing yourselves in some of the larger cities (Dublin, Belfast, Galway, Killarney, etc.) from which you can then book local day trips and tours as Anita advised.
There are express buses from the airport to several of the larger cities, as well as the Irish Rail train system which will get you where you want to go. The links provided by isn31c are all you need to get you started.
While having a car would provide maximum flexibility for exploring rural areas, you don't need (nor would you want) one in the cities. You can certainly enjoy a great first visit to the country without one.

My sister and I visited Ireland using public transport and had a lovely time. You can too!

You haven’t given any information about when you are traveling, your travel priorities, and how long you have for your trip. All of these pieces of information are helpful in getting more specific responses.

We had two weeks and like to spend 2-3 days in each location. That meant we chose to visit only certain places and did not try to do it all. We started in Dublin and then took the train to Belfast. From Belfast we took the train to Coleraine and transferred to a bus that dropped us off right at our hotel (Bayview Hotel, Portballintrae) and we explored the Antrim coast by foot and convenient bus. Next we took the bus to Derry where we transferred to a bus to Galway. A bit of a long travel day, but lots of time with locals and seeing small towns. Galway and Aran Islands were next. We also visited Ashford Castle and did the Hawk Walk. By then our 2 weeks were up and we took the train to Dublin where we spent the night and flew home the next day.

The key to Ireland by public transport is to limit your scope and not try to see all of Ireland and Northern Ireland at once in a short time period.

My wife and I are also fit seniors and will be going to Ireland next spring. We'll be basing in Galway with day trips to various places that interest us. We're been before, twice for me and once for my wife. We drove before but with age, we'll be taking a bus from Shannon. Everything needs a bit more planning but you can still have a wonderful trip.

The tran system works for getting to bigger cities, but the Bus Eireann network is more extensive. Our first trip, we immediately went to Galway (by bus), then worked our way south, focusing on the west side of Ireland. The Dingle Peninsula, and Dingle Town, aren’t reachable by train - go with the bus. There’s a running Irish Setter in the side, n stead of a Greyhound!

Cyn, you explored the Dingle Peninsula by bus? If so, how big was it and with who?