On weekends the beaches are very alive. Chinese discover the sea pleasures as did the French in the 1936. I swam in November, the sea was hot and clean (this is not allways the case).
Walk the little streets of the Island, admire the old palaces and rest under one of the big trees. Tourists shops abound and sell the usual stuff. Chinese tourists buy a lot of dried seafood, a specialty of Gulangyu. We were there on a Saturday. It was very crowded. But the tourists walk the same streets - where they find shops - quit the beaten paths and choose other itineraries. There are stone maps of the Island at many corners.
It's possible to listen to a 10 minutes concert on an old piano (ours was 80 years old) in the Museum. One of the ward staff did it. It was good.
Ask the staff to know when the next concert will take place.