purchasing unlocked phone in Zurich for use in Switz & Italy

Most European SIMs work in all European countries. It is a basic requirement. The EU regulates intra-EU tarifs so international calls within Europe are not very expensive.
You can buy a simple locked phone including a prepaid SIM card in most electronic shops ("Interdiscount" is the largest chain) for about CHF 20.
If you already have a compatible unlocked phone, you can buy a pre-paid SIM card for about CHF 15. For example, at Coop supermarkets with charges of 30Rp / minute for inland calls. There are many operators offerring "loss-leader" SIM cards. For example Lebara.
Just double-check when you buy your SIM which countries it covers and how you can top it up.
In Switzerland you have to provide ID (proof of name and address) when you buy a SIM card. The phone number is then registerted to you.

I would not expect fees from Italy to Switzerland to be very high, but check anyway as such fees are not regulated by EU - Switzerland is not part of EU.

Roaming rates are in general considerably more with a Swiss SIM compared with a EEA one, especially for data but even for basic telephony.

So, because we're landing first in Switzerland and buying a phone there, it will be more expensive to use in Italy since Switzerland isn't part of the EU? I guess it's not that big a deal since we're not planning to use it too much. We just want a cheap phone we can use in both countries for this trip. Thanks.

Once you take the Swiss phone chip into Italy, you will pay a higher rate per minute, but not as high as it used to be. Rather than trying to estimate how much money you'll need on the phone in Italy, I would just load on a moderate amount. If your time runs our while you're in Italy, and is not easy to top up, then you can cheaply buy a new chip there.

You could call your customer service for the plan you are using in the states & ask them how to unlock your phone. When you land in purchase a SIM card to use in Switzerland (removing & holding onto your USA SIM card for when you return). Purchase a SIM card in Italy once you are there & retain the Swiss one for future travels.

The basic idea to this comment came from an AT&T rep at my local store. You get to keep the phone you are using at home while paying a lower price for the calls & data you use.

Check with your phone service carrier.

Thank you all for your help.

$9 SIM from EuroBuzz.