Bocce?

In my observation, hit or miss. Maybe in a park or smaller town. The problem might be that if you run across a game, it will likely be older gentlemen, less english spoken, less interest in engaging you, might take some direct approach to make it happen. (Or organize yourself like a Rick Steves film crew and pull a fast one on them)

Agree with Paul. Very common in France; I didn't see it as much in Italy. Those who play seem like they are taking it very seriously. They may or may not be open to letting you join. Can't hurt to try! Enjoy your trip!

Yes, in smaller towns and mostly in the south. I would wait to be invited, but I am a shy person.

Saw some kids using the the bocce court in Monterosso last September

I know several places in Florence, but you can find most venues anywhere in Italy here:
http://www.campobocce.it/pages/view_page/1
There might be others 'less serious' elsewhere. And of course people play on the beach a lot too, but that's not the 'real' thing.

Serious players tend to be older Italians, as bocce is a sport particularly popular among the over-60 men. They take the game extremely seriously, so I don't know if they'll play with you unless you are really skilled. I've seen some of those old folks do stuff that I would not have the skill to even fathom. When I was growing up, my friends and I would spend hours in awe watching those grandpas play. Really fun to watch. Sometimes we would play too among ourselves, but we never had the skills to play with the old folks, they were at another level. The chance for older Italians to speak English is as near zero as you can get, so to interact you'll need to speak some Italian.

The only time I saw it in Italy was in the park near the Borghese Gallery (Rome). The players were as Roberto described. I don't know how serious they were, them seemed to be jovially chiding each other's play. They only spoke Italian, but they were friendly toward me. Maybe if I'd been inclined and had the time, they would have let me have a go at it.