Hello, while I do not live in Italy at present, my father had duel citizenship (he is now deceased, RIP) in the UK and USA, so I grew up in Europe--- I am reading this blog, with the intention of moving to Milan at some point, presently. I graduated from highschool in Germany and have lived in a number of other countries in Europe; what I can say, is that you probably can accomplish a great deal by internet, without ever setting foot into the overhead retail establishment of the bank.
You should try to search the internet and find the online websites of banks that interest you, first. Do some checking on them, see if they are reputable. And likewise, allow them to do checks on you, so their appraisal of you is not merely based on your fashion tendencies in a personal interview: but of your credit history and so forth. If you are confident in their professional standing, that is something that will greatly influence your satisfaction with their service.
Once you have established which bank you are interested in, if you want to have a personal interview, you might try calling the bank on the telephone, or emailing to set up the interview. In my experience in living in Europe, I found that bringing letters of recommendation with you, that have NOTHING to do with the bank, (or whatever you are visiting, its true of all beurocracies) really helps. So for example, if you have letters of recommendation from your own Embassy, or from your own foreign affairs office. Or, if you have letters of recommendation from the Italian foreign affairs office. Or, if you have letters of recommendation from your own employer, or even your own church.
In fact, if you do go to church, you may find that you are not permitted to receive communion in Italy without letters of confirmation from your Bishop, stating that you are in fact a member of said church. I don't know how that would work out in your case, since each denomination is different in how strict they are. The Eastern Orthodox church is very strict and will not allow new comers (even with the right papers) until after a 1 year sojourn "trial" (and no communion during all that time!) The catholic church may be less forbidding. Be sure to get such a letter, addressed from your rector to "whom it may concern" that you may present, as a testimony to your authenticity, if you wish to be able to receive communion in a foreign church.
Unlike churches, Banks have clerks which speed up the process considerably. And, your online credit report and various other public documents may very well make personal interviews less important. Try to be sure that you REALLY have the right bank, the first time, though. Switching banks is a pain in the neck. If you find that their website is completely "greek to you" (to coin a phrase) you might try downloading the free Google Translate Toolbar, that will translate all pages and all languages.