I'm an italian citizen, with a still valid ESTA. Anyway in the meantime I got a new passport, so I will have to ask for a new ESTA. Before doing anything wrong, I would like to know if I should do something with the old ESTA (like communicate to the embassy that it should be canceled) or I simply start a new ESTA request procedure?
Just apply for a new ESTA
From the ESTA FAQ :
Each Visa Waiver Program traveler must have an approved travel authorization for the passport they plan to use before they travel to the United States. If a traveler obtains a new passport, they must submit a new travel authorization application in ESTA using the new passport. A processing fee will be charged for each new application submitted.