Strategy for a Schengen visa application: Attending a course + a bit of tourism

As long as your main destination remains the same and the rest is incidental, you're fine, especially if you enter and leave through the country that issued the visa. Of course, you also need to respect the conditions of your visa (period of validity, duration of stay and number of entries).

Complications only arise if your travels suggest you haven't been forthcoming in your visa application and really had other intentions. Say you apply for a course in Italy and never go there but spend a long time in Germany. In that case, it is possible (although by no means certain, because enforcement is obviously difficult in practice) to see your visa annulled with all sort of associated unpleasant consequences.

One issue is that the visa you will get will be based on the evidence you submit. You might get lucky but for a first application, if all you have is a proof of registration from an Italian education institution, the consulate has no reason to allow you to stay much longer than the duration of the course. Simply asking for a longer visa likely won't be enough. If you are really planning a long tour and want to make sure you get a visa that allows that, you need to prepare an itinerary with as much supporting evidence as possible.

So you do have quite a bit of freedom but it's best to be as open as possible. If you know you will make a tour in Germany and Switzerland definitely mention that in your application but don't worry too much about Belgium. The key is to be honest and convincing, you can always remove a day there or add one here later on.

In practice, the borders between Schengen countries are very open and you can often cross them without seeing any else than a sign on the road. No stop, no passport check, no stamps. So a small change like driving a little further to spend one day in Luxembourg and Belgium when you are in Germany is not a problem at all.

See also Should my first trip be to the country which issued my Schengen Visa? for a discussion of a range of related scenarios.

I'm aware that the Schengen visa allows free movement across 29 countries in the Euro area, but how free is it, exactly?

As free as you want it to be. Once your Visa is granted, you can move freely across countries until your permission expires. Please note that even though the UK is in the EU, it is not participating in the Schenghen programme and, as such, if you wish to visit any part of it you will need a separate visa. Please see http://www.schengenvisainfo.com/schengen-visa-countries-list/ Border control between most countries has been lifted, so I wouldn't be suprised if you weren't even asked for a passport. That is not the case for air travel and Switzerland. (Ie: You will still be allowed through, but your documents will be inspected.)

Secondly, on the acceptance letter from Bocconi, the duration of the session is explicitly stated as 24 days. This is the only official letter that I have stating that they will support my stay in Italy. So I am in two minds about filling in the duration of my stay on the form. I want to extend it by a few days, just in case I want to roam around a bit more. What should be the way to go here?

The country where your trip begins decides whether of not to give grant the Visa. Your should contact the Italian Embassy in your country and inquire. Depending on the country you come from, you may be granted an extension. This varies and there is no global answer.

Thirdly, I want to know how long before the desired date of travel should I book my tickets for train travel in Italy and between Milan and Zurich/Frankfurt?

As soon as possible. It's international rail travel so prices tend to fluctuate. You may find air travel is actually cheaper. Check low-fare routes.

Any other travel advice is welcome. I am a 21 year old, crazy about travelling alone with little experience at that. This will be my first time in Europe and I will appreciate you guys doling out information that I might need.

Italy and Germany use the Euro, but Switzerland doesn't. Check conversion rates and card fees if you are travelling to there. Hostelworld and Hostelbookers have huge databases of cheap accommodation and you only pay a small deposit upfront, quite convenient. Most people in Milan do not speak English, including public transport operators.