Yes, you can move to Italy, but how long you will be able to stay depends on your situation. All Americans automatically receive a 90-day tourist visa upon entering the Schengen zone (which includes Italy). During this time, you can live and travel around Italy, exploring different regions and getting a sense of what life is really like. Note that while traveling as a tourist, you are technically not allowed to work.
After 90 days, you will need to exit the Schengen zone. If you would like to move to Italy and stay long-term, diverse opportunities for Americans, such as studying, retiring, investing, or working with a job offer. Research and consult to find the best path for you.