I am a non-EU student in Italy. Recently I have left my passport at the UK embassy for visa. However I have a ticket to for Greece in the next days (with Ryanair) and I the embassy doesn't give me a clear answer whether I will receive my passport by then. Now I would like to know whether I can travel to Greece with only my permesso di soggiorno and Carta d'identita Italiana (and no passport)?
Generally speaking travel documents for travelling within Schengen zone are: passport or EU/EEA national ID card. Officially none other documents are accepted, however some airlines might be more relaxed in their requirements. RyanAir is not one of these airlines, and RyanAir will not let you travel with just residency permit (I'm also speaking from personal experience).
Carta d'identita Italiana is not valid travel documents for non-EU citizens.