I traveled all over when I was there. I also traveled literally everything weekend and during the break.
Places I've Been and can offer an Opinion and advice:
Amsterdam, Netherlands
Amsterdam is a very small city. You can easily walk the city no problem without taking public transportation. I recommend the Van Gogh museum and of course walking through the Red Light District (no cameras allowed).
Athens, Greece
Lots of history here. It will probably take 2 days to do all the historical landmarks in Athens, and I recommend going to the Greek Islands if you have time. Just a note, Greece goes dead in October through Winter so try and go earlier if you can.
Barcelona, Spain
Late night party USA, I mean Europe. Parties start at 2am and can go til 8am, take a nap after dinner and rest up. Some cool clubs are Opium, Catwalk and Razzmataz. As for the tourist stuff, go check out all of Gaudi's stuff, the Olympic site, the beach and the churches. I spent a week here just cause it was relaxing, but you could cram everything into 2 or 3 days just depends on how well you work without sleep.
Bellagio/Lake Como, Italy
Absolutely beautiful, only way to describe it. Get on a bus from Lake Como and take it to Bellagio. Great place to go for a day trip. Train leaves from Codorna.
Brussels, Belgium
A really small city, can easily do everything in half of a day. Check out the Atomium, Manneken Pis (really much smaller than you think). Make sure to have a "real" Belgium waffle (not Liege), and a beer.
Cairo/Giza/Luxor, Egypt
If you have the money do it. When else are you going to be able to go Egypt for this cheap. Check out the Pyramids, Sphinx, and if you have a day check out Luxor (Valley of the Kings and Queens, Luxor and Karnac Temples). I would spend as much time here as possible. This could be a whole other site in itself, e-mail me if you want more info.
Dublin, Ireland
Possibly the drinking capital of Europe. Once you do St. Patrick Cathedral, the Guinness Brewery and Trinity College, it just turns into non-stop drinking at night. Everybody is just hammered at night. Definitely check out Temple Bar and everything is easily doable in a weekend.
Florence/Pisa, Italy
Florence is one of the art capitals of Italy. Check out David. The Duomo really looks fake is the best way to describe it. If you have time go horseback riding through the vineyards. Pisa is literally just a pass through. Literally have a layover there for 3 hours or go see the leaning tower and have lunch. Generally do it on the way back from Florence, Venice or Rome. A weekend should be sufficient.
London/Bath/Windsor, England
I recommend staying for about 4 days to get everything done. I recommend going half way to 2/3 into the semester or year pending on how long are you at Bocconi just because it will be nice to be around English speaking people. They have every type of food there as well. You do have to tip though. If you want to see more of England, I did a Greyhound Tour of Windsor, Bath and Stonehenge for about 60 US dollars (plus tickets) cause like Egypt when else are you going to be there.
PS WATCH OUT! Literally, make sure to look at the ground before you cross the street. It will tell you which way to look. You grew up look looking left, right, left when you cross a street or drive, that might get you killed in London.
Munich, Germany
One word: Oktoberfest. Whether you drink or not, you need to go if you're in Milan in Fall. It's definitely an experience and a great time. Registrar with IS@B for 125 euros for the bus ride and hotel. Else go camp out and rough it and make sure you have enough money to make it back to Italy when you're done. Just make sure if you want to leave a tent that you don't want to come back, cause you won't get in. Make sure to pace yourself, don't go too hard in the morning or you'll be out cold by noon.
Paris/Versailles, France
Some much to do, so little time. I could spend a week there easy. I went for 5 days and I didn't have enough time to do everything I wanted to do. You could easily spend days in each museum alone. Be aware of beggars at the Eiffel Tower, they are very persistent and annoying. We always got pastries in the morning cause they are also amazing. I also recommend spending a day Versailles, but make sure you go when there is a fountain show cause that's the best part.
Rome, Italy
A ton of history and a ton of places to go. The city is big but walkable. Try to figure out the bus system cause that will make it easier. The subway is great to get to the Vatican and the Spanish Steps. You'll probably spend a whole day at the Vatican and I recommend staying for about 4 days.
Turin, Italy
Probably my least favorite city of everywhere I went to. The city is walkable. There's really not much to do there, but it's a great place to go skiing or snowboarding in winter and check out the Olympic sites.
Venice, Italy
Exactly what you think of when you think of Venice. Gondolas are expensive cause they are a blatant tourist attraction. All public transit is water boats (sucks if you get sea sick like me). Go check out St Mark's Square, Basilica of San Marco, as well as the palace and the art museums. Definitely go to Murano during the day and watch the famous glass makers. The glass is inexpensive and make great gifts (just be careful not to break anything). I recommend staying for 2 days. I went for a day and it's doable but 2 days is best to be safe.