A very impressive entry! My corrections are below.

A vertical forest

Last Thursday I went to Milan for an interview. Sadly, it's been only a partial insuccess, but anyway I had the chance to visit that amazing city. I know, usually people come to Italy to visit historical sites, but I'm more into modern, innovative things.

Insuccess is not a word. We would say something was "only a partial success". Innovating is a verb, the adjective is innovative.

Milan has its relevant historical sites to visit - let's think at of the Cathedral ("Il Duomo di Milano"), or at the Sforza Castle ("Il Castello Sforzesco") - but what I really admire there are the new buildings, particularly skyscrapers ("i grattacieli", or "le torri", as we call them).

"Let's think of" is a little akward. An English speaker would understand what you mean, but we wouldn't normally use that phrase. Maybe "Milan has its relevant historical sites to visit - I'm thinking of . . . " You also need a stronger punctuation mark than a comma here because the two parts of the sentence could be their own sentences. I would use a dash.

Most of them are very recent or still under construction, and they rise so quickly that, using an Italian idiom, we can say "crescono come funghi" ("they grow like mushrooms").

Wonderful!

Italians usually are not likely to accept innovative architectural shapes in general, but, in my opinion, all our marvellous historical sites have been built at a certain point in the past. And usually they went against the grain, at first.

Great use of an idiom here.

Now we should preserve them more carefully, of course, but why just live in the past? Why should we stop building great sites?

In my opinion, in Milan they seem to understand this point, and that new quarter, Porta Nuova ("new gate"), is a new architectural marvel. Those skyscrapers are fantastic, so high, so nicely colored, their shapes being amusing and harmonious.

"Amusing" normally means funny. I don't think we would call buildings amusing. Interesting maybe.

I don't feel oppressed by those giants, their sight brings a sense of joy to me.
Even their names are nice: "Il Dritto" (the straight one, but "dritto" also means a cunning person), "lo Storto" (the twisted one), "Il Curvo" (the curved one) are some examples.
But there's a building complex in particular which I wanted to see.
It's name is "Bosco Verticale" (vertical forest). It is made of two towers rising from a green area. Their peculiarity is that they are entirely covered by trees. Their surface is green and alive. Trees convey fresh air and acoustic isolation to them, together with all the moral benefits that plants give to humans.

Moral benefits? I'm not sure what you mean by this.

I had my lunch (a "calzone") sitting in the grass while gazing at them, and I felt like a child at Disneyland.
I read that they are building a "vertical forest" in Sidney, too. That's great. In my hope this is how cities will look like in the future: green parks and monumental buildings blended together.

"How cities will look" or "what cities will look like" not "how cities will look like".

Amazing job!

http://giardininelmondo.it/il-bosco-verticale.html