Considering a trip for next year and am thinking about Ireland. Would like to stay 5 nights in Dublin and 4 nights in another area, perhaps spending a 10th night near either Shannon or Dublin airport for the return flight. We don't want to rent a car but would consider hiring a car and driver for some excursions so a bigger town might fit the bill. We like beautiful scenery, museums, shopping for local goods. We like Irish music (DH's family emigrated from Ireland to the US in the 19th century), tea, scones, fine Irish butter and simple fresh food. Where would you send us for a second place to stay?

Belfast or Galway

Kinsale too

Kinsale would require a car and driver. Killarney, Galway and Belfast all have very good day tour options and connect to Dublin easily via train or bus.
Without knowing your interests, it's hard to recommend specific bases.

so many travelers ignore Dublin, while in several visits I consider it one of the most underrated cities of its size in Europe. Five days will be perfect for you...even toss in a day trip in the vicinity.

For second location, I agree with our knowledgeable Italian Chauffeur. Interact with the locals...that is the real charm of Ireland. Trad music is readily available in all the above locations.

Galway