27 day 1st time travel to Ireland

With your interest in history, you really need to go to Newgrange. There are day trip tours from Dublin.

Save Dublin for the end of your trip. After originally landing in Dublin, take the next Citylink bus from the airport to Galway. It’s my favorite city in Ireland. We got front row seats on the upper deck of the bus, got a mini tour of Dublin as it passed through enroute to Galway, got a nap during part of the four hour ride, and were dropped off close to our Petra House B&B. The beaches just west of Galway, in Salthill, are nice. We stayed in a B&B in Salthill once, too, but I prefer being in Galway.

From Galway, visit the Aran Islands, or at least Inishmore, the largest. You might not choose to stay many nights, but it’s a good short-to-moderate stay. Get there by ferry from Rosssveal, or better yet by small Aer Arran plane from the small Connemara Airport.

Visiting in May will let you stay in Ennis during the Fleadh Nua music festival, the best in Ireland, if you’re interested. We were there for several days and nights two years ago, and it was outstanding.

On three trips, we’re traveled mostly by bicycle (the first trip), or rental car, so other than the Citilink bus and some regional buses, I don’t have tons of public transportation experience in Ireland. There are some trains, and but they don't cover everywhere, although Galway and Ennis are served by trains.

We’ve split trips into being either in the west, north, or the south of Ireland. On our last trip (south) two years ago, we stayed awhile in Macroom, a place Rick’s book says is good for a lunch stop between Cork and Kenmare, but it was good for longer! Again, we had a car, so I can’t offer how practical it would be for using public transport, or maybe has organized taxis. An amazing, unexpected place we discovered for nature and hiking was the Gougane Barra, but I can’t say how to get there without a car.

Dingle is one of Rick’s best “back doors” in Europe, and we took Bus Eireann regional buses between Tralee and Dingle to get there and back. Their buses have a running Irish Setter on the sides, instead of a Greyhound like I was used to in the USA. So, Galway, Aran Island(s), Ennis, Dingle, then up to Dublin, maybe by way of Macroom, Cork, and/or Kinsale.

Sounds like a wonderful trip! A very helpful, reliable website for transportation is https://www.seat61.com.
Dingle is our favorite but not the easiest to reach by public transportation. There are plenty of good day trips from Dublin, Galway and Belfast using tour groups; RS recommends some. So choosing a couple of home bases and taking day trips from those is a good option.

Have you thought about adding a couple Rabbies Day Tours to your itinerary? They have two based in Killarney., one features Dingle and the other is Ring Of Kerry. I won't have personal experience until mid-July, after your trip. I signed up for both, plus two days tours out of London. The company has been recommended several times on the forum. https://www.rabbies.com/en

Since you asked for itinerary help, hopefully this will give you a couple ideas. For my week in Ireland, I'm flying into Dublin after almost three weeks in England. As soon as I land, I'm taking FlightLink bus to Killarney for one free day to either go to the horses races or go riding. Then each of the next two days are the Rabbies day tours. The following day I head to Dublin to sightsee on my own and visit friends. Itinerary is not finalized yet but includes National Museum of Archaeology, Dublin Castle/Chester Beatty Collection, James Distillery tour, Kilmainham Gaol and Book of Kells. Depending of friends' schedules, I spend two days with them. But if it's only one day, my plans are either a day trip to Kildare/Irish National Stud or see a couple small museums in Dublin.

Last visit to Ireland was 33 years ago, so I'm excited to be returning. On that trip we really liked the Burren, Connemara National Park and Dingle’s Gallarus Oratory.
Sláinte!

The Irish National Stud is great! You get to see the foals running around and I really enjoyed the tour. Definitely recommend that one to anyone who likes horses.

The RS guidebook should have some good itinerary suggestions, including a public transportation option. Rick's usual advice for making an itinerary is to list everything you want to see and do, determine how much time each takes and eliminate as needed. The trains run out of Dublin and Belfast like spokes on a wheel. Getting between cities, especially in the West, is by bus. If you need specific connection information, ask isn31c, who is likely to comment here shortly. That fella (I assume) has the Irish transport system at his fingertips. The most sensible it to do a loop, either counterclockwise (e.g., Belfast, Bushmills, Derry, Killybegs/Donegal, Westport, Galway, Dingle, Ring of Kerry, Cork/Cobh, Kilkenny, Glendalough, Dublin) or the reverse. With nearly a month, you shouldn't have to compromise on what to see.

If you use the search feature on this website you will find the question of Ireland by public transit has been asked many times, and you will find some great advice in those posts. I will say that Killarney, Galway, Belfast, and Dublin were the four larger towns I used as bases to do day trips from. Some smaller, more remote places, such as Dingle, are difficult and time consuming to get to by public transit, but a guided day trip will get you there. Since you have already booked your flights in and out of Dublin, you may indeed want to look at taking a bus to one of the other towns to start your journey, and ultimately finish in Dublin. The first thing to do is to figure out what is most important on your list of things to see, and plan your route around that, and try not to backtrack too much.

Further to @ G3rryCee's thoughts I am trying not to comment where we don't have an itinerary yet. That seems like putting the cart before the horse to me. But @ G3rryCee's outline itinerary is not a bad one. Although like many such on here it doesn't give much time to the South East of Ireland- the likes of County Waterford and County Wexford

But 3 initial points-

a. @Cyn's thoughts on Macroom- if you are coming from Dingle towards Cork then Macroom is on the hourly bus route #40 Tralee to Cork, so that is not a problem.

b. For some reason it is often said or implied on here how hard Dingle is to get to by public transport, implying that it is easy by car. However you look at it or get there Dingle is a long way west. Get to Tralee on the train (which is not much longer than driving, at least from Dublin) and you then have buses. Or one other option from Dublin is domestic flights to Kerry airport. By the time you have got to DUB, hung around for a flight etc I'm highly unconvinced how much time you really save over the train, and I am seriously not in favour of such short domestic flights (in Ireland, the US or anywhere- and those who know me mean I can also be a hypocrite on that score domestically in the US and Scotland)

c. Due to the geography of the West Coast Donegal to Westport is an awkward if not impossible journey. Be aware of the twice daily Donegal to Dublin flights then the Dublin Heuston to Westport trains. Or there is a surprisingly good Donegal to Dublin Airport and Dublin City bus service. Another half way compromise is bus to Sligo (overnight there, I know- a short stay), then next day train (or bus) to Dublin [change rail stations using the LUAS tram], train to Westport.

Unlike many people you have the luxury of almost a month, so are not fighting to fit a quart into a pint pot- or a lot into a week or two weeks.

To me you need an itinerary first then sort out the transport, as opposed to having a transport driven itinerary. Given the pace at which public transport is changing and improving in Ireland it is possible that even Rick's 2025 book may be out of date before it hit the streets. For instance Spine Route 6 in Dublin has just started this week- without going into detail that has utterly changed things much for the better between Dublin and Dun Laoghaire, Bray and south thereof down the east coast.

I agree it would be a challenge to get from Donegal to Westport in a day. Particularly if one has not planned well and is traveling on a Sunday, as happened to me last summer. Thankfully, we had planned an overnight in Sligo, but it took us all day to get even that far. For Yeats fans, Sligo merits a couple of days. Also, the Carrowmore Megalithic Cemetery is there. I had hoped to see that while we were there, but it wasn't to be.
Honestly, I didn't mean for my list of cities to be considered an itinerary suggestion. My main suggestion was the circular route, rather than going back/forth to Dublin by train.
One place that's been on my list for several years is Athlone, which is on the train line, and offers several interesting sights, both in the city and nearby. If one were cutting out the North (which I don't recommend) or putting Dublin in the middle (which isn't a bad idea, making roughly a southern and a northern loop), one could continue on to Westport or Galway from there.
The strategic use of taxis is another way to make traveling by public transport easier. For example, last summer we used taxis to get into the Sperrins Mountains (pre-booked) and to visit my friend's family sites in and around a small village in Mayo (this was done on the fly by hiring a taxi in the nearest town with a taxi service).

Here is our plan so far:

1. Land at Dublin airport spend 4 night. Two days walking around and enjoying Dublin and one day taking day trip to Wicklow mountains and Glendalough
2. Train to either Cobh or Kinsale for 3 night stay - day in Kinsale a day in Cobh and maybe day trip to Rock of Cashel
3. Bus to Dingle for 4 night stay one day to bus tour Ring of Kerry and 2 days walking around Dingle and Peninsula.
4. Bus to Doolin for maybe a night but for sure a ferry to Aran Island for 2 nights.
5. Ferry to Galway for 3 nights. Day trips from here to different locations along coast and inland.
6. Bus to Derry for 4 nights day trips to Donnegal and Giant’s Causeway.
7. Bus to Belfast for 2 nights.
8. Bus to Drogheda for 2 nights to day trip to Newgrange.
9. Bus or train to Airport. We need to add one night someplace. We use Road2Rio for setting up buses and taxi where needed. We’ll try to arrange train from home to save money. Please make and comments, suggestions or questions you have to us. Thanks for caring.

We found bus travel difficult between towns. If possible I would avoid using Dublin Express or the Big Green bus in any of your travels. We had two very bad experiences with them.

There are lots of buses from Derry to Belfast, by several routes. But there is also the hourly train service to consider.
It is a bit unusual here to get someone preferring bus over train when they are equally as good.
From Belfast to Drogheda there is no direct bus service- you would have to change at least once, whereas there is the hourly direct train service (the Enterprise to Dublin).
From Drogheda to the airport the direct bus is the obvious choice as against train into Dublin then bus back to the airport.

We are using Expressway busses for some of our trip this Spring. My son, who lives in Ireland, uses these busses often for longer journeys.

In Drogheda, stay at the Marcy Hotel. Central location, walking distance to the bus station to get the bus to the airport.
You might add a night in Sligo between Galway and Derry. Also, I would shift one night to Belfast and do 3 in Derry, 3 in Belfast and do the Giant's Causeway from Belfast with Paddywagon. If you don't add the stop between Galway and Derry, add it to Dublin. There is really so much to see in the city. Actually, I might skip Drogheda and do Newgrange as a day trip from Dublin. End the trip in Belfast.
Google is better than Rome2Rio, in my experience, but Transport for Ireland is better than either. Direct messaging isn31c, if available, is superior to all others.

I know Rick Steve’s doesn’t like Killarney, but don’t discount it too quickly. I spent four nights there and really enjoyed it. Doing a day trip that included crossing the lake in a wooden dinghy and then a jaunting car ride through the Gap of Dunloe was one of my favourite experiences.

Dave and Mary, when you say a train to either Cobh or Kinsale for 3 night stay, make that stay in Kinsale, a much nicer town for those 3 nights. A day trip to Cobh from Kinsale will be better than a daytrip to Kinsale from Cobh. Kinsale’s a bit farther south, but it’s a far more pleasant setting.