Galway day trips?

Will you have a car? There are loads of things to see in Connemara but best if you have your own transportation IMO.

You don't specify how hardy the cyclists in your group are, so I'll offer this. Bike the Connemara loop on the N59 (or sections of it), in my experience the most beautiful region in Ireland.

Head west from Galway and go through Oughterard, Maam Cross, Clifden, Letterfrack, Leenane, Oughterard again and back to Galway. Probably the most scenic and peaceful piece of this pie is on the north-south N344 though the Inagh Valley between Recess and Kylemore Abbey. Absolutely stunning.

In Leenane, have dinner at what once was called the Blackberry Restaurant (and before that the Field). Next door is the new Purple Door Cafe, great for a light breakfast or lunch.

Per Katheryne, you really need a car to see the best of Connemara.
EuropCar might be your best bet for a rental but you'll want to check their operating hours before committing to anything. They do appear to have vehicles with automatic transmissions in their inventory ... which is what you want.
A very nice route from Galway would be to head clockwise along the N59 towards Clifden, with maybe a stop there to stretch your legs and explore the pretty little market town. Just west of Clifden is Sky Road - one of Ireland's great scenic drives on a sunny day - which features spectacular seascapes from every bend in the road. It's one of those very narrow (single lane in places) rural roads that Ireland is famous for, but it's worth it for the views.
Further along the N59 is Connemara National Park, which has nice little hike up to a local feature called Diamond Hill. From the top you'll be treated to picture postcard views of the surrounding hills and the sea off in the distance, and on the way up you'll encounter some of the wild goats that populate the area. The visitor center has a little tea room where you can grab a coffee, cup of tea, or a snack to fortify you for the rest of the morning.
Continuing on you'll find Kylemore Abbey. Can google it to see what it's all about.
If you're adventurous you can continue north up to the village of Leenaun. Located on Killary Harbor, it's Ireland's only true Fjord.
A particularly evocative place a bit further on is the Doolough Memorial just north of the village of Delphi. You can google that too to read about what happened there during the famine years. The place looks like it could be in the Scottish Highlands - worth a stop if you're interested in a crash course in a bit of Irish history.
The village of Cong could be a stop on your way back towards Galway town, though I think it's worth a day to itself. There's a very popular Hawk Walk at Ashford Castle that many here rave about, and having lunch or tea afterwards would make for a memorable day. Cong is where the old John Wayne film "The Quiet Man" was filmed back in the early 50's and the village still looks like the movie set it was back in the day.

I spent four nights in Galway without a car a few years ago, and I just took some guided day trips. One was to the Ciffs of Moher and Burren, another was the Connemara area that included a stop at Kylemore Abbey, and the third was a trip out to Inishmore which was the bus ride to and from Galway to the ferry, and the ferry ride over. Once on the island I was on my own for the day. If you don’t want to rent a car the guided day trips work well.
I hope the Crane Pub is still open. They have a room upstairs called the listening room where they bring in some of Irelands best musicians. It’s a small room, and you have to buy tickets, but I listened to some great music there.

Ditto on The Crane Bar. We're always there in April/May and tickets weren't required upstairs. There's music downstairs, as well. There are may good pubs on that side of the river (Beirhaus, Carroll's) with music on some days, and there's several more on the mall (An Pucan, Tigh Coilli) that have music, but they seem to be more touristy.

I'd suggest an overnight on Inish Mor if you can stick it on one end so you don't pay for lodging twice.

You may not be as excited about music as we are, but Mannions, Lowry's, and EJKings in Clifden are good pubs.

The Great Western Greenway near Westport might also be of interest - I wish I wasn't a broken down 73 year old.

A lot of people find Galway boring, but we don't. It's actually quite a small, walkable town (with sprawling suburbs).

I looked up what you are seeing on your RS tour and I agree that the Aran Islands are a good day trip, but an even better overnight. We enjoyed Ashford Castle (Cong) and the Ireland School of Falconry. It was possible to get there by public transport but it takes some creativity. PM me if you want more info.