My wife is a Galwegian - well, Connemara - so I’ve spent a lot of time there over the years. First piece of advice: it might be the height of summer, but the weather will be shit for all or part of your visit. Cold Atlantic winds bringing rain will hit you at some time. Bring at least a fleece and a rainproof coat.
Galway city’s delights are many and obvious, so I shan’t go into detail, but I’ll tell you what you can do nearby. Galway city sits on the eastern end of Galway Bay, so from the city you’ve got two real choices (east Galway is IMO dull as ditchwater) - south west towards Clare, and north west into Connamara.
You can get to the Burren in Clare in about an hour. The Burren is a vast, bizarre geological phenomenon with its own ecosystem that I haven’t the energy or knowledge to describe, but it’s well worth a visit. Then a little further south to the Cliffs of Moher, which are the most spectacular cliffs I’ve ever seen.
Going the other way, I’d recommend a drive along the north of Galway Bay, turning right at Ballynahown to Maam Cross, then west again to Clifden, which is a lovely little town, albeit very touristy. The inland of Connemara is like nowhere else I’ve ever seen - blasted stone and red heath scrub as far as the eye can see. The first time I went there I described it as like being on the surface of Mars. On your way back, there’s a fantastic restaurant at Moycullen, called Moycullen House, which does fabulous “modern Irish cuisine”. It’s a bit expensive and you’ll need to make a booking, but it’s well worth it.
The area is firmly “Gaeltacht”, which means that most people speak Irish there as their first language. However, with the exception of some very elderly people, everyone also speaks English fluently. Don’t be insulted, as some people have been, if someone speaking English to you reverts to Irish immediately when talking to others - they’re not being rude or talking about you behind your back, it’s their native tongue, and they were doing you a favour. We had a cute experience there once when we went to get a sandwich - I ordered in my English accent, so the sandwich shop people didn’t know my wife could speak Irish. The guy barked something to the woman in Irish, and I felt a trifle paranoid. However, my wife recounted later that the fella had said “Mary! For God’s sake put more ham in that sandwich! What will those nice people think of us with you being so mean?”
If you have time I’d also recommend a trip to the Aran Islands, which you can get to by ferry from Galway, or a ten-minute hop in the smallest plane in the universe from Inverin, where they weigh you before getting on. The Aran Islands are amazing looking, and home to tiny isolated communities completely out of time, seemingly having moved little since the nineteenth century.
Most of the pubs around Connemara will have live music of some kind each evening, just happening spontaneously. In Galway city it’s sometimes staged for the tourists, and sometimes it’s actually a booked artist playing, so joining in wouldn’t always be appropriate. However in Connemara they genuinely appreciate the “session” as an artform rather than a tourist attraction.
Most people will absolutely love it if you join in - and spectators will voice their approval of particularly good bits of playing with the words “mahu” (not sure of the spelling of these words) or “fair play”. And “ahrís” (ah-reesh) means “encore”.
However, there is a subtle etiquette, that I’m not really tuned into. I was once playing guitar and singing in a pub in Spiddal, and a load of people came in with violins, sat in another room and started squalling away - they were individually very good but en masse they were deafeningly loud, sounded hideous, and totally drowned me out - but after about half an hour the landlady came over and told me to stop playing as I was “putting the fiddlers off”.
It’s a lovely place, most people are truly as warm and welcoming as the clichés say, and were it not for the awful weather and the lack of jobs, I would have much rather lived there rather than ten years in Dublin. Enjoy!