Whether you are religious or not, you can not miss a visit to the shrine of Santa Rosalia, located on top of Mount Pellegrino (which among other things Goethe had called the most beautiful promontory in the world).
The cult of Santa Rosalia in Palermo is sacred, being the “Santuzza” patroness of the city. The sanctuary is located inside a cave, where Rosalia lived the last years of his life.
If you really want to do something by real Palermo, then do not miss the acchianata, or the ascent on foot of the mountain on the old road. Mystical, yes, but also invigorating!