Winter Ireland Itinerary Help - no car with 3 people

In addition to the info about public transport in the Rick Steves guide, you might find this site useful

https://www.tripsavvy.com/around-ireland-without-a-car-1542965

If I were you I would probably stay in Dublin for the first couple of nights, if not longer, to let the jetlag wear off and get used to Irish ways of doing things in a reasonably cosmopolitan city. Then when you return, do spend at least one night in Dublin so you don't have a lengthy distance to travel to get to the airport on time.

In setting priorities for where to go and how to spend your time, think about why you chose Ireland in the first place. Do any of you have family roots in certain counties/towns? Are there certain places in Ireland you've always wanted to see, or certain Irish experiences you've always wanted to have?

Dingle is a must, IMO.

For Day trips out of Killarney

Consider Dingle / Ring of Kerry / West Cork all very scenic

Consider private Tours

Try Ring Of Kerry Sightseeing Tours https://www.facebook.com/ringofkerrysightseeingtours/

I agree with Eric that Dingle is not to be missed!! Also, we visited Cobh and found the museum there so full of history regarding the Irish immigrants. Tipperary was our least favorite stop but if you have friends there that may change things.

Too bad you're not renting a car. Driving on the right is not difficult and in winter you won't have tour buses to swerve around. Best adventures we had were getting "lost" then discovering out of the say things, like a 300 year old woolen mill (I can't tell your where because we just followed signed and ended up there). The Burren in NW Co. Clare has very interesting rock formations and stone age burial monuments, but again, you'd need a car. Walking around Connemara Park with it's Connemara pony herd was interesting, even in the rain. Cliften's a good village to stay in. I'd avoid the tourist spots like Bunratty. I wasn't impressed the Galway city, nothing quaint about it. The worst "Irish" music I heard was at Ashford Castle, catered to tourists who don't visit pubs. Instead stop by any pub on most nights and enjoy the locals. Cong Abbey near Ashford Castle is a well kept ruin, but I wish I could remember where the fantastic out of the way ruined monastery with crosses peaking out of windows was, one of those signs we followed. Gallarus Oratory on Dingle Peninsula is a must see. Visit Waterford Crystal to watch the blowers and cutters and view the amazing showroom chandeliers, but don't expect discounts. If you collect Waterford Christmas ornaments like I do, you may find some that aren't imported to your home town.
I'm guessing from your moniker that you're a horse person. When we were in Ireland, Ashford Castle was still using the "Quiet Man" set for its riding stable. There are places to ride, like Castle Leslie, but do a Google search for stables around the country and trail rides on the beach. Stop at the Irish National Stud in Kildare to see their Thoroughbreds and their horse museum.
Enjoy your trip. Sláinte

I have done Ireland without a car, and it is doable, but realize that you will need to base yourself in towns that will have some mode of transportation to get you to the places you want to see. That may be public transit, or that may be guided day tours of some sort. Just know that when you rely on public transit it will take you a lot longer to get where you are going so you won't be able to see as much in a day. For this reason I did not stay in Dingle during my trip as it was going to take up too much of my valuable touring time to get to and from Dingle by public transit, and at the time there was not a tour company operating out of Dingle that would have allowed me to see the sights I wanted to see. I stayed in Killarney instead and did a tour from there. So, I would suggest you make a list of where you would like to stay and then check to see how difficult it is to get there, and then whether or not there are touring options that will get you around once you are there. Some towns have regular bus service from a major train station and are easy to get to, others require getting to a certain point by train then taking a couple of buses. I would not use Cork as a base as it's not a very interesting town for a longer stay. Kinsale is only an hour away by bus, and that bus runs once an hour. Killarney, Galway and Dublin are the larger towns that have lots of options for days tours, but public transit in Galway and Killarney is not great. I would suggest that you look at a map of the train system first and start your planning from there before you get your heart set on staying in a certain town and then realize it will take half a day to get there by bus.