Western Scotland tends to be a bit warmed and wetter than the east. You can check actual, day-by-day, historical weather statistics at timeanddate.com. Look at several years' worth of data to get an idea of the conditions you may encounter. I'm cold-natured and stayed warm during July 2019, despite often being caught in the rain, by wearing merino long johns and 97%-nylon slacks that were reasonably effective at shedding water (though not waterproof). I had a waterproof, hooded jacket and waterproof shoes. I also used an umbrella. My major rain issues were in Oban and on Skye.
Glasgow's weather for June 2021: https://www.timeanddate.com/weather/uk/glasgow/historic?month=6&year=2021
Glasgow has a number of good museums, including the Hunterian Gallery, Hunterian Museum, and Kelvingrove Art Gallery and Museum. There are others as well. There's a botanic garden that would be nice in good weather.
Glasgow is famous for its Charles Rennie Mackintosh buildings. Unfortunately, the Glasgow School of Art has suffered two catastrophic fires in recent years and is a long way from reopening (if it ever does). If you're interested in Mackintosh architecture you can get an overview at The Lighthouse, which used to do walking tours (book ahead). I tracked down most of his local buildings and had a great time doing so. Three of the best are the House for an Art Lover, the reconstructed home at the Hunterian Art Gallery, and Hill House. Hill House is in Helensburgh, about an hour from Glasgow by train then a decent walk from the most convenient RR station (I'd guess taxis are available; don't know about city buses).
Both the House for an Art Lover and Hill House had 2-for-1 deals available at the time of my visit. Both parties had to have rail tickets to qualify. Maybe you could use a return ticket from Helensburgh to qualify at the House for an Art Lover??
My shopping is limited to contemporary crafts, so this list may not help you. I haven't checked to see whether these places are still open; even in normal times, craft galleries don't necessarily have staying power.
In the center:
- Scottish Design Exchange, 220 Buchanan St, MC: Probably lower-end. https://www.scottishdesignexchange.com/contact
- The Roger Billcliffe Gallery, 134 Blythswood St, at Sauchiehall Ln: Heavy on art but also has some glass and other crafts. Website suggests calling first 0141 332 4027. http://www.billcliffegallery.com/
- ScotlandArt Gallery, 193 Bath St, just E of Douglas St: Art, jewelry, a few ceramics. https://www.scotlandart.com/
- Mary Mary, 51 Oswald St: Art. Tue-Sat 1100-1800. https://www.marymarygallery.co.uk/
- Koppe Astner, Suite 1-2, 6 Dixon St: High-end temp. exhibitions, might occasionally include glass or ceramics. https://koppeastner.com/
- Briggait: Historic market building now with artists’ studios. 141 Bridgegate, Townhead.
- Nancy Smillie Jewelry Studio, 425 Great Western Rd. https://www.nancysmillieshop.com/
- Thistle Gallery Scottish Contemporary Art, 56 Park Rd, 1 block S of Great Western Rd. Art and crafts, including glass. https://www.thistle-gallery.com/
- Nancy Smillie, 53 Cresswell St, Hillhead: shops with Scottish products, including ceramics. https://www.nancysmillieshop.
- Annan Gallery, 164 Woodlands Rd, just W of West End Pkwy: Art and crafts, including glass by Scotland Glassblowing. http://www.annanart.com