Carrying placards bearing the name of famous books, thousands of protesters marched through the streets, some wearing helmets similar to those used by riot police.
As some protesters began trying to smash the windows of a stationary police van, one student tried to reason with them not to use violence.
There were not yet any confirmed reports of injuries to police or protesters in Palermo.
In Rome, police were out in force following violence at a demonstration last week, during which protesters torched cars, smashed shop front windows and destroyed bank ATMs in the historic city centre.
Thousands of students marched through the streets and along Rome’s main highways, causing traffic to come to a standstill.
“If they block our future, we will block the city,” sang some students as they marched.
One of Rome’s most famous fountains on Gianicolo Hill ran red after demonstrators poured red paint into the water.
Students said the red colour symbolised both the blood spilt in last week’s violent protests, and the budgets of Italy’s education system.
Cleaners had already begun to drain the fountain and said that no permanent damage had been done.
Many of the students in Rome who had begun marching early on Wednesday morning made it clear that they intended it to be a peaceful demonstration.
Some carried flags and banners emblazoned with non-violent slogans, and many had painted their hands white as a representation of their peaceful intent.
Forty-one people were taken into custody and almost 100 were reported injured in last week’s protests.
The ANSA news agency reported that protests in Rome, Milan and Sardegna had so far passed off peacefully.