Students planned a protest in a city suburb where a 16-year-old, the son of a union leader, was wounded in a shooting on Wednesday night.
The students also planned to stage a concert at Athens University.
The main trade unions organised separate rallies outside parliament against an "anti-workers" budget due for a government vote on Sunday.
After Thursday's violence in Athens, th opposition Pasok socialist party again called for early elections, arguing that the government had lost control of the situation.
About 100 leading figures, including academics, magistrates and economists, launched a petition calling on the government to re-establish public confidence.
Athens police said they had arrested eight youth after several hours of street battles with dozens of protesters in the city centre on Thursday.
The youth had broken off from a peaceful demonstration of about 5,000 to confront police officers.
During the course of Thursday's clashes three cars, a Greek flag and tables and chairs from neighbourhood cafes were burned.