Galway

I've been to Galway twice.

See Galway Cathedral. Opened in 1965, it's considered the last great stone cathedral of Europe. The Celtic design is stunning.

Take a tour -- if one is offered -- of the Claddagh, the town that gave rise to the eponymous ring. I've never been on a Claddagh tour, but there must be one. I walked through the Claddagh on my own. Also, tours of the Burren district and the Aran Islands can be booked in the Galway tourist information office.

At atmosphere neighborhood is called Nun's Island. Visit the house of Nora Barnacle, the spouse of James Joyce. I had bad luck: Twice I stopped by the house, and both times it was closed! (The proprietor of one of my B&Bs told me that her mother had known Nora and that her family disapproved of Joyce because he was a sophisticated Dublin man and writer. The proprietor told me that Nora would go out in the street to let off steam and carry on.)

If you're there in July, go to the Galway International Arts Festival. I've always wanted to see it, but haven't. I'm told it's not just another arts festival and that it's pretty amazing: https://www.giaf.ie/festival/events

The best Irish music performance I ever attended was in a Galway pub, but I can't remember the name! Locals came in with the instruments and started playing. People spontaneously got up and started dancing Irish style. It was better than any of the 50 or so Irish music concerts I have been to in arts centers. Ask around: Locals will tell you.

The atmosphere of Galway is the best in Ireland. Take a walk. A nice walk is to Salthill, a beautiful tourist town with stunning views of the Atlantic Ocean: https://www.galwaytourism.ie/towns/salthill/. That's also a great place to stay.

Historical walking tour by the Galway Civic Trust.
Walk or bus to Salthill.
Day trip to Connemara.
Galway arts festival -- the official festival and the fringe festival offer music, art, crafts, theatre, food in July.
Trad music at local bars.

Check out the Crane Bar http://www.thecranebar.com/ I heard the best music of my trip there, and any Celtic musicians i know always go there when they are in town.

Frankly, just walking through the old town is a trip. It's no surprise that Galway is one of the most touristy places on earth. Beautiful place, though.

"... Galway is one of the most touristy places on earth."
And therein lies the problem - for us anyway. It's never been one of our particular favorites, though if you'll be without a car it would make a good base for day trips around the area as has been mentioned.
If you do plan to have a car then consider the pretty (and relatively uncongested) little market town of Clifden about an hour to the NW of Galway. It sits in the center of the beautiful loop drive, the Sky Road scenic drive which starts just west of the town, and is only about 15 minutes from the ferry landing at Cleggan from which you can take the short boat ride over to one of Ireland's best kept secrets - the island of Inishbofin.