Unlocking cell phones

shows you how awful At&T is. anyway https://www.att.com/deviceunlock/#/

AT&T is about the same as any company having a phone under contract, basically you need an off contract phone, one you replaced, or you can buy them online. Relying on someone over there to hack your phone may cause problems once you try to get back on AT&T. I would maybe look for a phone to use, then if you can get by using your iPhone on international roaming ($1.25/min, no data) for other things, go with that.

You can add their basic international plan for $30 for a month, and keep your phone number by doing so.

I am under contract with Verizon. All Verizon LTE phones/ devices are unlocked. I can suspend my account and purchase a Italian SIM with no problems at all. AT&T does not do the same?

I bought a TIM phone in Italy a couple of trips ago (replaced an older one), cost less than €50 including the minutes I bought to use. I suspended my Verizon account while I was away.

"AT&T does not do the same?"

No, and neither does T-Mobile. As said above, their phones are sold locked, and they will only unlock them once you have paid off the full phone price (which for an iPhone can be about $600 - or even more, if you have the 128 GB model). T-Mobile does this without a hassle (I've done it for several); a few years ago AT&T was reported to give more trouble, but this may have changed.

I don't recall seeing unlocking places in Italy, as phones there are sold unlocked; of course, this doesn't mean they don't exist. I did see them in France where phones are sold locked, but I was last there in 2010, and someone here posted that those places have since disappeared (don't remember details).

I was under the impression that verizon phones were locked from the factory. Maybe wrong about that. If you have a Verizon phone, it uses LTE. Isn't that incompatible with the European GSM, or does the unlocking process allow you to use the phone anywhere in the world.

Verizon LTE phones are unlocked and Global Ready, meaning they have CDMA and GSM radios inside.

I stand corrected on the above. I did some checking and verizon lte phones are unlocked, You just need to buy a sim card when you arrive in Italy. The store will usually put it in and you should be good to go.

I would take Zoe's suggestion and buy a phone in Italy. Less hassle than trying to work with phone companies in the U.S. We have had many friends visit who have thought that they had their phones ready to work in Europe, only to find that they wouldn't work at all.