Sicily has a long-standing tradition of pastry making whose heritage was left by Arabs, Greeks and also Spaniards. That's one of the reasons of which Sicilian desserts are so rich, diverse and commonly made with sweet ricotta and almond paste. I particularly love two of them: the cassata and the cannolo.
Basically, the most interesting restaurants, bars and pizzerias in Palermo are in the Old Town Center, known as Borgo Vecchio. There are other very nice places outside this area, but I recommend exploring this region that has a huge historic and cultural richness.