A smart phone may not cost much more than a flip phone and is infinitely more useful when traveling - e.g. maps and navigation, better messaging, VOIP calling (e.g. Google Hangouts, Skype), etc. But it depends what your primary purpose is in buying the phones: to keep in touch with each other if separated? Calling locally? Calling home?
Buying Android phones at home before getting to Italy may make some sense; you'll have time to turn the phone on and setup contacts while on WiFi, etc. You won't have to spend as much time in Italy figuring out how to use them. I'm not sure what to recommend right now for cheap Android phones. I just bought an unlocked Moto E4 for $105 on sale, but it's going to be my primary US phone (will work with any US carrier) - and should work great in Europe with a SIM. You can probably find cheaper phones than that. Often they go on sale but you must make sure the phone has the right frequencies to work in Europe. You can buy used unlocked phones on eBay (and Amazon) if you know what to look for. You can probably buy flip phones that will work too. My ancient 10 year old flip phone should still work in Europe though it's long since stopped being useful to me.
If you own smart phones already, they may work in Europe if unlocked. If they aren't unlocked, unlocking them may be relatively easy. (Verizon smart phones aren't locked.)