I personally use DeGiro. I just saw they operate in Italy:
https://www.degiro.it/I don't think you can buy funds, but they have a list of 200+ ETF without commission, plus no yearly maintenance fees. Probably not the best option for trading, since the interface is pretty basic, but it does work pretty well for buy & hold purposes. I do have my taxable account there, with no issues so far. I buy the World, EM and SC ETF and had no issue so far. You can buy ETFs from almost every big player: Vanguard, iShares, Amundi, etc.
Another popular option is Interactive Brokers, they do charge a 10 EUR/month fee from which they discount the commissions monthly. I do not personally use it, so I cannot recommend; but from what I've heard is very good if you want to do some more advanced investements.