Cotton or linen only in hot weather. Not that thick fruit of the loom stuff but some good fabric that is thin enough to breathe, but it's still cotton so it's wicking moisture. Evaporation results in a cooling effect so long as the cotton is thin enough. That's the only way to go in the heat (imo). Cheap old men's dress shirts from the 80s that are still 100% cotton and have been washed too many times.

Merino wool t-shirts are pretty great in hot weather.

I’ve never tried running in them, but lightweight wool socks (e.g. darntough) are pretty great for summer wear. Last a lot longer than cotton too.

I use wool socks year round, even during summer for hiking or general purpose except at the gym. They are excellent for rains too.