Public Transport to Cliffs of Moher and Coast from Dublin (Temple Bar area)?

You understand the Cliffs of Moher aren’t a hop, skip and a jump from Dublin. In fact on the opposite side of the island.

Cannot advise about public transport, I drove.

And by the way mostly all of Eire is beautiful.

Why not take a tour from Dublin to nearby Glendalough and Wicklow Mountains? Much easier.

Probably easiest for you to take the train to Galway (less than 3 hrs.), spend a couple of nights, and book day trips to attractions in the area from there - the Cliffs, Burren, Connemara Loop, etc. The Aran Islands ferry company has a collection point in Galway too, so you could consider that as well.
Lots to see and do in the area. Worth 3 or 4 days IMO.

I agree with Robert, just take the train to Galway and book a day trip from there. The day trips will probably make a stop in the Burren area also, and maybe somewhere else. There’s lot of day trips you can do from Galway so it might be worth a couple of days or even more depending on where else you want to go.

So, now that I've looked up Galway....I am thinking we could definitely spend much more time there - just so much to choose from!!! Could anyone tell me if it's realistic to be able to do day trips to Cliffs, Connemara National Park and potentially seeing the Aran islands - all without a rental car? Also, would Aran islands still be a good choice at the end of May?

I agree with the others about staying a couple of nights at least in Galway, I loved it. As for day tours that you mentioned, all would be available as organized tours (1/2 day or full day, single destinations or combos) from Galway. Independently by public transportation is not particularly convenient so a rental car would be best if you don't want to join an organized tour.

Yes to all of your questions.
Depending upon the weather on the day, May would be fine. Can check the forecast the day before you commit - the ferry doesn't sell out.

Yes, you can do all of those day trips without a car. It’s exactly what I did. I spent 4 nights in Galway. I came by train from Kilkenny so my first day wasn’t a full day so I used it to wander the town. Then over the next three days I did guided day trips to the Cliffs/Burren, Connemara/Kylemore Abbey, and then out to Inishmore on my last day. It was great fun!

You don't need to do a guided trip to Aran Islands unless you want to. The ferry company has a shuttle that goes from downtown Galway to the ferry dock. You can take the shuttle and ferry and arrive as independent travelers if you like.