I would be interested as well and have been to roughly the same places as you have. Not too many years ago I went to the cathedral not expecting much, but was surprised in that it was quite grand for a small city like Turku. Other places I recall visiting are Forum Marinum with its Tall Ships and a museum of musical instruments (can't recall the name).

I'm perhaps a little more familiar with the surrounding archipelago which is definitively recommendable especially for sailing in the summer.