The town has an assortment of religious buildings, some quite old. Only one Mosque of the many that used to exist survives, but it is worth seeing. The Qingjing Mosque is on Tumen Jie. There are Taoist, Buddhist, and Confucian temples, as anywhere in China, plus Christian churches. One large and impressive Taoist temple is just east of the mosque. There are also Hindu and Zoroastrian temples, and the world's only surviving Manichean temple.
The "Old Saint", an enormous statue of Lao Tse, the founder of Taoism, on Qingyuanshan just outside town attracts people from all over China.
Quanzhou is famous for puppets. There is an excellent free puppet museum. They sometimes do shows, which are excellent, but not on a regular schedule. You need to be lucky to catch one, or to have a group of 20 or so people and make arrangements.
Other museums include the Fujian-Taiwan Kinship Museum, the Quanzhou Museum, and the Maritime Museum.