My English experience.

English uses the article (that is, the word "the") a lot less than Italian does. In English you would usually only use "the" when you are referring to something specific. For example, "I studied at the Napoli High School".

In this case you aren't naming the school, just saying that you studied at a high school level.

Similarly you use capitals only if you are giving the proper name of something. For example, the Naopli High School. When you are talking about high schools in general, you don't use capitals.

You could also say "at high school"; that would be perfectly correct. However I've changed it to "in high school" since I think it's a little more common.

I attended a scientific high school where two foreign languages were teached taught: English and French.

Not all English words add -ed to form the past tense. A lot do (shopped, looked, walked, etc), but many don't (swam for swim, drove for drive, etc). In this case "taught" is the past form of "to teach".

I used to like studing studying new languages at school. Then I graduated and I started the university.

Again we aren't talking about a specific university (say, the University of Bologna), but rather about you studying at university level. Therefore there is no "the", and the U is not capitalised.

I decided to study computer science, but in the curriculum there is only a little English exam that you have to do on the computer. It's a sort of multiple choice test, with easy questions.

"Sort of" doesn't add anything to the sentence. The test is a multiple choice one, so you can say that without qualification.

So I have taken it but I haven't improved my English much. Now I would like to start speaking English again also because next year I will probably have some university courses in English.

That's very good. You clearly have a well developed understanding of essential English vocabulary and grammar.