I'll need to give this Italian phone number to everyone back home who may need to contact me while I'm gone, right? (With a warning that it will be an international cell for them)
Correct
All Italian cell numbers have a 3 digit area code starting with 3 (the 330's are assigned to TIM, while the 340's are assigned to Vodafone for example). The number must be preceded by Italy's code, which is 39. For example your Italian cell phone number might be something like:
+ 39 (333) 555 5555 (Tim)
+ 39 (347) 555 5555 (Vodafone)
The + is the international outbound code. From a cell phone you just need to dial a +, but from a land line in the US the + sign stands for 011, which must be dialed before the rest of the number.
The call from the US will be a call to an international cell phone, which is more expensive than calling an international land line (my long distance carrier charges me US$ 0.36 per minute to call an Italian cell phone from the US). Let your family be aware of that price when they call you, so that they don't stay on the phone too long. If you have a tablet or smart phone it is a good idea to use Skype while connected to the hotel wifi for talking to the US, that would be free.
What happens to the voicemails and texts that go to my original number? I'll probably want to check them a couple times.... Do I put my original SIM card back in for a few moments to do this?
Yes.
That will be international calling then?
No.
Since you are calling your voicemail (a US number) from your US cell number. However you will pay international roaming charges which range from US$ 0.99 to 1.29 per minute for most US carriers.
Note:
Using your Italian SIM card is simple, just ask the store to install for you. Also ask to explain how to access the voice messages left on your Italian number. Not sure if the voice message prompts come in Italian only or if you can change the language to English. In any case you just have to know two digits: the digit to press for "play the message" and the digit to press to "erase the message". The store can provide that for you and any Italian who uses the same carrier (for example Vodafone) can tell you anything you need, the prompts are the same if they have the same provider. For example I can never remember what I have to do with my Vodafone from one year to the next, but many of my relatives have Vodafone, so I just ask them. For example to know how much money you have left in your sim card before having to fill it again, just dial 904 and Vodafone will send you an SMS with the amount left.