Ireland trip recs

I'm thinking about a week in Ireland. The tentative plan is to head straight to Galway on an express bus from Dublin airport. I would spend three nights in Galway exploring the city but mainly using it as a base for taking guided tours to Connemara region and

Then I would take the train to Dublin city center and spend the remaining three days exploring all of the main Dublin tourist spots plus a day trip to Kilkenny. Morning flight home out of Dublin.

Questions:

(1) any tips for things I must do over there - sights to visit, things to do, places to eat or sleep?

(2) I'm thinking of extending my trip by a day or two and visiting a place other than Galway and Dublin. I do not want to rent a car (solo traveler, never driven on other side), so anywhere I go has to be accessible by train or bus and fit within a mid-range budget and timeframe. Any suggestions? I'm curious about Killarney, Sligo and Athlone but will consider anything cool.