Sims card

I don't have Verizon therefore I will let others comment on the pros and cons of Verizon international plans. I will only give you information on the local Italian SIM.

To buy a local SIM in Italy you will need an unlocked smart phone. If you purchased your phone through Verizon and had it for less than 2 years, it may not be eligible for unlocking. Check with Verizon in that case. You will need the unlocking code number from Verizon in order to be able to work with a SIM from another provider. If you purchased your phone on your own, for example at the Apple Store, then the phone is unlocked.
To buy a SIM card, when you are in Italy, simply go to a Vodafone or any other provider store or any electronics store that is an authorized dealer (most are) and purchase a local plan. For foreign visitors I recommend Vodafone Holiday plan. It costs 30€ all inclusive for a month. Details are below. Remember that in Europe you pay only for calls or texts you send (outgoing), receiving calls and texts is always free.
http://www.vodafone.it/portal/Privati/Tariffe-e-Prodotti/Tariffe/Estero/Vodafone-Holiday-English
The other major provider, TIM, has a similar plan called TIM Welcome.

The pros of such local plan is that you get a lot of data and free international calls and texts for less money that you would spend with most international plans, including Verizon.
Your smart phone will have the exact same capabilities you have now.

The cons are that your phone will now have an Italian number. Therefore if your family wants to reach you you have to communicate to them your new Italian number. Italy area code is 39, all cell phones have mobile specific area codes that start with 3. Vodafone area codes are in the 340s. So for example for your family to reach you they might need to call a number such as
+ 39 346 555-5555
If they call from a land phone instead of the + sign they would dial 011 first, then 39 etc.
Since it's not cheap to call an Italian phone number from the US, generally over 30 cents a minute, if they contact you, it is cheaper if you call them right back using the free international minutes you get with the above Holiday plan. And of course if your phone has already apps like Skype or Whatsup or Vibe, or FaceTime (if u have an apple device) you can use those features, which use data instead of minutes.

To use an Italian SIM, you will need a tri-band or Quad-band GSM phone. Verizon, by default, uses CDMA cellular networks - not GSM. However, they also sell "world phones" which contain support for both CDMA and GSM. All iPhones since the 4S also contain both technologies. As far as iPhones go, if you have a 5S or newer, it is already "unlocked" for foreign SIMs per an agreement with the FCC. Just remember to remove the SIM upon return because your phone will not re-connect to the Verizon network if the foreign SIM still installed. (I can't speak to any of Verizon's other phones because I'm not familiar with them.)

As for blue tooth and WIFI, it is advisable to keep them turned off whenever not being used. However, don't be afraid the use local WIFI, when available. Just don't use open WIFI to conduct personal business (banking, etc.) or your personal information may become compromised.

Another option is to buy or rent a mobile hotspot and just use your phone's WIFI capabilities...

There is no need at all to turn off your wifi. If I want to avoid using data, I put my phone in airplane mode, then turn my wifi ON. It will function whenever there is wifi available (which can be quite a number of places). In your case, I would get an international plan from Verizon simply because you want to keep your same phone number so it's easy for your family to call. An Italian SIM gives you a different number. You can still keep your data off most of the time if you want. Generally, I don't care about calling minutes, but I could use more data.