Congratulations Josh on your wedding. I just returned from Ireland (skipped Northern Ireland this time), staying in Dublin and Galway and then day trips. I agree with the recommendation to take the bus to Galway and start your trip there. It will save you time backtracking to Dublin to catch your flight home.
For Galway, we stayed at the Harbour Hotel. It is good location, about 8 minute walk from Eyre Square, has air conditioning which we needed in July and you may need in August and has parking (which is a bonus if you are driving). The place for trad music in Galway is the Crane and it typically starts around 9:30 p.m. The upstairs is a music venue with well known acts and usually needs to be booked in advance. The downstairs you can come and listen, just get there early to have a seat. Since you will be on the WestEnd of Galway, if you want to have a wonderful dinner before the performance, I can highly recommend Kai. This is a Michelin Bib Gourmand restaurant run by a local chef Jess Murphy. All the ingredients are local, but there is a modern take on the cooking. It is a bit pricey, but the food is amazing, especially the seafood. Kirwan is also a great place for seafood and the famous Galway Oysters. Both places require reservations.
In Dublin, I can second the recommendation for the Brazen Head. If you don’t mind grabbing a taxi, the Gravediggers is also a fantastic pub situated right next to a cemetery. It is off the beaten path, but excellent. Also for trad music in Dublin, can recommend the Traditional Irish Music Pub Crawl hosted by two musicians. You visit two different bars and listen to great music and learn about the instruments and the songs. The other recommendation is Cobblestone pub. We learned about the pub from our friends whose family lives in Dublin and it is more traditional than some of the music we heard in Temple Bar.
For the Guinness Brewery, recommend that you get the experience with the Stoutie. They take your photo and print it on a head of Guinness and it is remarkably clear. You will still get your Guinness pour at the Gravity bar, but this experience is more fun and certainly less crowded.