Best International bank account

It depends where you are going to be living.   If you have a credit agricole in your town, then they seem a good bet.  There arent many intl banks in Italy so your choice is  Deutschebank, a couple of French ones and perhaps a couple of others.  None of which have country wide coverage, and tend to be very localised.  If you're in MIlan you will have more choice.    If you mean a bank whcih can deal internationally then some of the Italian banks are good, but expensive.   

@Modicasa Thank you for your response.  I want a bank that will accept UK and Italian payments into my current account.  I have no idea if this is possible.

Any bank will accept payments from third countries, but as you are receiving GBP presumably and wanting to deposit that in euros will end up costing you alot with traditional banking.  If you get a Wise card you can transfer your money into euros and then into your italian bank as a direct SEPA which costs nothing.  Im sure there are loads of other methods.