Visiting Ireland in June - where to go?

Hi Ask Ireland,

We'll be visiting Ireland the first week of June for a wedding in West Clare/Spanish Point. We plan on spending the wedding weekend there seeing the Cliffs of Moher and the Aran Islands during our free time.

Our question is - where to next? We will be flying in and out of Shannon, so we are OK with dedicating our remaining 4 - 5 days on the west coast. Should we stay in Galway and do day trips from there? Should we hire a driver to take us around the island? Any cities and gems that are absolutely worth seeing that aren't on the typical tourist radar? Any and all tips are welcome :) We are so excited to finally visit Ireland!!

Thanks in advance!