All recent and popular smartphones work internationally and every major US carrier has international plans. That means that your smartphone should automatically work once you arrive in Italy. However, some of these international plans are expensive and data roaming may be costly and slow.
An economical option is to get an Italian SIM card. Please check with your company to see if your phone brand and model will accept an Italian SIM. Most American phones are electronically locked so that one cannot switch SIM cards. But it is possible to get your phone unlocked (call your company), allowing you to replace the original SIM card with a local Italian SIM. Please note that in most cases the unlocking must be done while in the United States.
The most reliable Italian mobile-phone companies are TIM, Vodafone, and Wind. They have stores in Milan Stazione Centrale. As a foreigner you may find prepaid plans that offer 50GB data and unlimited calls and text (only to Italian numbers) for as low as €20 per month. However, when purchasing one of them you must consider that:
1) You must buy your Italian SIM in person and you must show your passport.
2) Besides the charge for the first month, when you open your account you must pay a fee for the SIM card and add a few euros to keep your account in the black. If at any moment you have zero or negative credit, your account will be blocked and you will not be able to make calls or use your data even if you still have many unused local minutes or data.
3) Some plans may include a certain number of minutes worldwide; otherwise international calls are charged per minute and paid with your credit. Incoming international calls are not charged to you. WhatsApp is an excellent way to make international calls at no cost (you need an internet connection).
4) Before the end of the first month, you must remember to top up your credit to cover the automatic renewal of your plan.
6) Credit is sold by most companies online and usually can be paid with PayPal but not necessarily with foreign credit cards. You can also go to stores and ask for a ricarica (particularly tabaccherie, newsstands, or a store of your cell phone company).
6) IMPORTANT: Unfortunately, by just viewing certain web pages you might be automatically redirected to a different site or find a background tab opened to a link that you didn’t request, and the simple access to that page could start an unsolicited weekly subscription to mobile services of games, music, or adult content, immediately paid with the credit available on your phone account. When you become aware of this charge you can call the customer service of your carrier to deactivate the subscriptions but all charges that already occurred will not be reimbursed. In order to prevent these charges to occur, call your carrier immediately after the opening of the account and activate the “barring SMS” (blocking of all premium rate services). The numbers of the major carriers are: TIM: 119;  Vodafone: 190;  Wind: 155. Note that in this way you will block also services that you activated intentionally.

Please be aware that calls from Italy to US toll-free numbers may not route, depending on the originating carrier. If they do route, then the caller would need to pay international dialing charges, just like for any other call to a US local phone number. If you expect to be calling your bank or another company or institution, ask them to provide an alternate US local number for international calls.