SIM cards in Greece and Italy

Any SIM card bought in any EU country (including Italy or Greece) will work in any other EU country (including Italy or Greece).
Unlikely to be coverage on the ferry, once it gets out of range of land, especially inside a metal ship.

A new SIM card means a new (Greek or Italian) phone number, you will have to exchange phone numbers.
One option is one of these companies (not recommending, just informing you they exist), the may be able to send you SIM cards before you leave home.
https://lebara.com
https://www.lycamobile.it/en/

You need to check:

• Are the phones unlocked?
• Do they have SIM card slots in them (presumably with SIM cards in them already)?
• Do the chargers work on European electricity (220V 50 Hz AC), and do you have enough plug adapters for the chargers?

Why do college students, who are presumably all over 18 and adults, need chaperones ?

If you want to buy a SIM ahead of time for EU countries, another option besides the two mentioned above is the Dutch Vodafone SIM:

https://community.ricksteves.com/travel-forum/tech-tips/dutch-vodafone-sim-card-for-use-in-europe

Many EU countries (like Italy and Greece) require you to register a SIM in country with your passport, but some countries like the Netherlands and the UK do not, so you can buy SIMs from companies that operate in those countries ahead of time before you get to Greece.

But if you don't mind waiting until you get to Greece, you can certainly buy SIMs as soon as you arrive:

http://prepaid-data-sim-card.wikia.com/wiki/Greece

All of those phones must be unlocked ahead of time in order to install another SIM in them.

If the old flip phone isn't a GSM phone (in the US - T-Mobile or AT&T), leave it home - a Verizon or Sprint flip phone most likely won't work at all in Europe. See if the old flip phone even has a SIM card; if not, forget it. If it is GSM, it must also be unlocked. And...even then it may not have the right frequencies to work in Europe anyway. You can find its model number and google its specs to find out. (The smart phones are more likely to be able to work in Europe though.)

If you happen to be from the US, install the Google Hangouts app on your smart phones, and you can make free voice calls to US phone numbers on WiFi or with mobile data, even to landlines (to call the airline or whatever). If everyone in the group has Hangouts, you can communicate without phone numbers, using Hangouts. Hangouts works to non-US numbers just not for free - a few cents/minute, like Skype. And if everyone is from the US, know that both T-Mobile and Sprint have free data roaming abroad, so those phones wouldn't even need SIM cards to make calls with Hangouts. AT&T and Verizon have $10 per-day international roaming plans.

I wouldn't expect mobile coverage while on the ferry unless you are close to land. When I took the ferry from Helsinki to St. Petersburg a few years ago, I was able to pick up mobile signals occasionally from Russia or Finland here or there, but we weren't too far from land. The ferry might have WiFi but probably not free.