Agree with ChateauMargaux and UpsilonPi's recommendations
Definitely the Burren and the islands over the Cliffs of Moher.
Galway is nice.
All of the Connemara coast and the Inagh Valley are gorgeous. Letterfrack and Achill lovely,
Killary Harbour ditto.
Many lovely towns in Mayo ( Westport, etc).
Ballycroy National Park/Nephin very wild and unspoiled and often wet and windy
Sligo and Yeats country well worth a visit.
Glenveagh National park in Donegal...
I would head out to Glendalough from your base in Dublin, and Powerscourt is really nice.
Howth to the north of Dublin has stunning views of Dublin, the bay, the mountains. Great cliff walk.
You could take a tour boat into Dublin Bay.
If you get nice weather, there are some lovely beaches in Dublin.
The Dart southbound to Bray followed by the Bray to Greystones cliff walk is nice. The Dart runs along the cliffs in the Dalkey/ Killiney stretch, with a view you may recall if you saw the movie 'Once'.
In the south east, Kilkenny is worth a visit - bustling medieval town, castle. The south of County Carlow around StMullins has lovely trails in the nearby Backstairs Mtns, canoe hire on the river Barrow, some charming towns - Graiguenamanagh, Thomastown ( in Co Kilkenny), Bunclody across the mountains in Wexford .. New Ross and Enniscorthy are nice too.
Beaches at Fethard and Baginbun in Wexford might suit.
Rock of Cashel in Tipperary might interest you.
The Dingle peninsula is usually heaving with tourists all summer, but the Beara peninsula isn't as well known. West cork in general is superb.