Cell Phone Use in Italy for one who does not use cell phones much???

First question I think you need to ask yourself, would you want the sitter and others calling you at all? If so (reminding them of the time difference - when they get home from school you will be asleep), would you prefer it if they call a US cell number, your existing number, or would you want them calling an Italian number, dialing internationally?

In other words, is familiarity and simplicity more important or cost?

A few thoughts......

1. You'll have to talk to the Virgin Mobile rep's about that. Another question to ask them is whether your present phone has the necessary frequency bands to operate in Europe. It's quite likely that your current phone is a CDMA device which doesn't use a SIM card and therefore is NOT compatible with the GSM system used in Europe and much of the rest of the world.
2. Based on your situation, my suggestion would be to buy a phone from one of the travel phone firms such as iRoam, Cellular Abroad, Telestial or Mobal. These often use a U.K.-based number, but have consistent rates throughout much of Europe. Many used post-paid billing (calls charged to a credit card) so there's never any need to top-up. Choosing a firm that offers an optional U.S. number in your home area linked to your travel phone would be a definite advantage!
3. NO, it's not necessary to buy a SIM card in Italy for it to work there. In the past I've used either a travel SIM or my roaming with my home cell network, and both have worked just fine.

The travel phone firms mentioned above also sell inexpensive phones or packages with a phone and a SIM card. Some of the plans can be renewed from year-to-year, so can be used for future trips as well.

Our long term cell was/is a prepaid Virgin Mobile flip phone. Cheap especially if you don't use it, but no roaming even in the States in that it only works through the Sprint network. I believe what they call their "International Plan" is intended for use in the States for calling certain countries outside of the US, most commonly other North American countries. I suspect that the Virgin Mobile flip phone will not operate in Italy. (We recently bought a smart phone with Cricket plan as our primary cell service but the VM prepaid is so cheap, we are keeping it a a second cell phone).

https://www.virginmobileusa.com/vmu/pages/international_connect/irates/irates.html?ID16=spot6:All:international%20use
Use your Virgin Mobile phone to dial internationally while in the U.S. International Long Distance calling capabilities may be enabled on your phone at activation.

Our foreign travel keep in touch in an emergency strategy is E-mail and leaving a detailed itinerary with overnight locations and phone numbers. Given the 8 to 10 hour time zone differences, E-mail is a practical message drop. (We have been fortunate not having an emergency situation where someone needed to be in immediate contact).

You may want to consider some sort of limited power of attorney to address medical needs for your dependent daughter. (We leave a son's contact information with our pet boarding kennel).

Thank you all so very much for your prompt replies!!
I had planned to leave a detailed itinerary with contact numbers and we most likely will have a way to keep in touch via email.
I think the information on getting a simple phone with prepaid plan and SIM card or whatever makes sense, and I had already come across one of the phone company such sites. This is such a great site to get firsthand,quick and complete answers to questions.

Leaving a detailed itinerary with contact numbers is effective to some extent, but there could be a delay of several hours if you happen to be out touring for the day. Using E-mail also has some inherent delays.

Although you mentioned "no texting", one other benefit of a cell phone is that you'll be able to receive updates that all is well at home on a regular basis, and in many cases received texts are FREE.