making calls and staying connected while Italy

Check out tourist SIMs from Vodaphone and TIM. They are the top networks in Italy and offer data/call cards specifically for visitors. Here's a link to a website with more detailed info.

"what is the best way to stay connected while traveling in Italy?"

This depends on what you mean by "best," and what you mean by "connected," and other specifics. Which country are you from? Does your current phone plan work in Italy, and how much will it cost to work there? Is your current phone unlocked and able to accept an Italian SIM?

Are you looking for data, calls, texts? Just to home, or just to your travel companions, or to Italian numbers too? Do you want or need to receive calls from your home country? If you want data, how much will you need in addition to hotel Wi-Fi - just a bit for maps, or a lot for other applications? How long is your trip?

Etc., etc. Much like any "best" question, there is no one answer for everyone.

Having said all that, if your phone is unlocked, look at TIM for Visitors (data only, 5 GB for 30 days for €20): https://www.tim.it/offerte/mobile/estero/dallitalia/tim-for-visitor-uk

and TIM Welcome (calls and data, 100 minutes and 4 GB for 30 days for €20): https://www.tim.it/offerte/mobile/estero/dallitalia/tim-welcome-internet-uk

if you are just going on vacation for a couple of weeks email works fine. you can connect using email or putting stuff on Facebook
but then again not everyone has access to the internet

Have a smart phone or a tablet? Use the Google Hangouts app to call home to the US for free, even to landlines. Use WiFi if you don't have mobile service in Italy. You can buy a SIM in Italy if your phone is unlocked. AT&T phones that aren't paid off won't be unlocked; Verizon phones aren't locked. T-Mobile and Sprint users get free international roaming data, free texting, and 20 cents/minute calls.

We tell our (grown) kids and the dogsitter to send us emails, which we can read using hotel Wi-Fi. Given the time difference between Europe and California, it's easier than trying to schedule when both sides might be available to talk or text. (And cheaper, too!)

"it's easier than trying to schedule when both sides might be available to talk or text"

I don't find texts to be a problem, as the recipient will read them when convenient (and when they're awake). I've found that to be the best way of keeping in touch with family at home.

If you have an unlocked smart phone you can purchase and install a local SIM card.

Be aware that if you buy a local SIM card, your phone will have an Italian phone number, therefore if people want to contact you, they'll need to call that number. however your Whatsapp, Skype, FaceTime, and other apps are unaffected, therefore people will be able to reach out to you through those. But if they call you on your phone, they need to call the Italian number. Since some plans offer international minutes included, it's cheaper if you call them, rather than call you. Most American long distance companies charge 30 to 40 cents a minute to call a mobile phone in Europe.

The best plans for visitors are Vodafone Holiday and TIM for visitors (Vodafone and TIM are the top two providers):
http://www.vodafone.it/portal/Privati/Tariffe-e-Prodotti/Tariffe/Estero/Vodafone-Holiday-English
https://www.tim.it/tim-visitors-en

Both are similar and both cost 30 euro for the 4 weeks.
TIM gives more data (4Gb) Vodafone more international calls and texts (300 of both) but only 2Gb of data.
I usually get Vodafone, but only because my country home in Tuscany is in a spot that gets very little TIM service.
I also get a SIM card for my iPad with Vodafone. That gives me the ability to use my iPad anywhere, even in the middle of a forest. SIM card + 10 Gb was 20 euro last month for my iPad.

With an American mobile phone provider you will pay more, however T-Mobile has good plans at decent prices if you have T-Mobile, I'd probably go with there plan which offers unlimited data worldwide. If you have an iPad with SIM card capability also, I would just buy that one from Vodafone or Tim.