To travel from the U.S to Italy or the U.K
If you're fully immunised you won't need to take a test in the USA before departure, just a lateral flow test (day zero to two) on arrival in England. You'll need to book and pay for the test before departure and use the information to complete a Passenger Locator Form, which will be verified by airline staff at the airport of departure: https://www.gov.uk/guidance/travel-to-england-from-another-country-during-coronavirus-covid-19
Are you travelling to Italy from the UK or as a separate trip from the USA to Italy? When do you plan to travel?