Would an MBA be useful for me??

Prima di tutto, congratulationi ad essere a Bocconi. Congratulatzioni sul GMAT anche, mamma mia! In confronto a la tua domanda.... e un po difficile dir' di si or di no ....

That is, a Masters in Financial Engineering will prepare you very well for a career in quantitative finance and trading. In my experience, those with an MFE tend to go more into the software development side of things - working on arbitrage programs, etc, but I'm sure someone here can speak ot this better than I can. In other words, an Masters in Finance can get you to the sales and trading desk of a major firm, yes.

An MBA from Harvard or Stanford would also do the trick. I'd also point out that it doesnt have to be from Harvard or Stanford - don't be so arrogant so as to presume you will choose from one of these two - you may very well not be (though harvard does like the young applicants). In any case, if you do pursue an MBA I'd suggest you look a little bit beyond Harvard and Stanford. Wharton and Chicago are just as well placed (and maybe even better) for a career in Sales and Trading. Stern and Columbia as well would likely do very well in this area. I know you are at the Harvard of Italy right now but .... All I'm trying to say is look beyond rank, and look at which schools truly excel in this area.

So, as to your question - can an MBA get you on a prop/trading desk or get you into investment management / wealth management on either the buy side or sell side?

Absolutely. It is a common path, and often with many of the firms you seek. For instance see: https://www.chicagogsb.edu/corp/hire/employmentreports/

As to which one will serve you better --- I think you need to do a few things.

1) Start by comparing placement statistics. I know Bocconi is very respected within Europe, and especially within Italy, but I really dont know
how well it does in recruiting. That is, does Goldman come to campus? Does Lehman? At top MBA programs, you will find all the top investment banks.
2) Decide if you want to work in the US or in Europe. Although a good percentage of people pursuing an MBA in the US do move to Europe or Asia for their jobs, a large percentage do not. Odds are, if you pursue an MBA in the US, you will stay in the US.
3) Consider long term prestige. I seriously considered applying to European MBA programs - including INSEAD/IESE etc ... and even Bocconi - they keep calling me in fact - but I decided not to. The reason was that I came to realize that a top MBA education in the US carried almost as much weight in Europe. That is, recruiters in Europe knew the names of top MBA programs in the US. I found that, to some extent, people in the US were less familiar with european names. But you really should be deciding based on #1 and #2 .

So, without knowing how strong Bocconi is in this area, I don't know which is better for you. I do know that a top MBA program can get you into the career path you seek.

Bouna fortuna amico mio!