I'm a permanent resident in CA holding a green card but not US citizen. Do I need to pay taxes on interests earned (capital gains on stocks/CDs) in a foreign country (Italy in the specific)? Also, how becoming a US citizen would change things in these regards?

Yes, as a California resident. You must claim ALL world-wide income on your California return, just like you claim such world-wide income on your U.S. tax return.

No, becoming a U.S. citizen will NOT change this requirement.