What is the school system in Italy like? From what age do you start study? What are the names of the grades, or forms, or years? Is it free? Do you have to pass special tests to finish?

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there's a preschool system that goes from 0 to 3 and from 3 to 5-6 years old, but it's not compulsory.
Actual school starts at 6 and until 16 students have to go to school.
elementary school (elementari): 6 to 10-11 y.o., 5 years
middle school (medie): 11 to 14 y.o., 3 years
high school (superiori): 14 to 16-17 or 18-19 y.o., 3 or 5 years.
high school is divided into: istituto professionale, where you learn a job, and usually last 3 years; istituto tecnico, where you study more theory stuff but you are also prepared for a job, usually last 5 years; liceo, you only study, it doesn't prepare you for a job, lasts 5 years, and it's recommended to go to university afterwards.
There are various kinds of high school: for example you can choose one where you learn how to work with wood, or another on how to be an electrician, or you can study economy and law stuff, or languages, science, maths.... There are many.
university (università): lasts 3, 5, or even 10 years, depending on what you study and when you decide to take your exams.

every time you change school (from elementary to middle school, to highschool) the grades are counted:
1st year: prima
2nd year: seconda
3rd year: terza
4th year: quarta
5th year: quinta

in uni, you'd say:
1st year: primo anno
2nd year: terzo anno
3rd year: quarto anno
etc etc

you have to pass a special exam at the end of the last year of middle school to be admitted into high school and another exam (called maturità) at the end of the last year of high school to prove that you finished it.

School starts in September (most universities in october) and finishes in June (for unis I don't really know). We have holidays on the 1st of November, on in the 8th of December, two weeks for Christmas, some days for Carnival, a week or so for Easter, on April 25th, on may 1st and in June 2nd.

Apart from university, education should be free, but every year you have to pay an "enrolment tax". It isn't too expensive, usually less than 50€. Starting from middle school books are to be bought by students and those are frickin expensive, if one wants to buy them new, it costs at least around 200€ every year. And school trips or other activities are financed by students since schools don't have money.
There are also private schools and those may offer some more services but obviously they cost more.

University isn't free and a public one costs around 700-1000€ per year.