I've seen a few of Rick's shows on Italy and when people are out for the evening or at dinner, they seem to wear a variety of jackets. Is a sport jacket is needed to fit in?
As in the US, that will depend on your restaurant choice for dinner, but in most cases no. Wear what is comfortable.
A lot of Italian men will throw a sportcoat or blazer or equivalent over what they're wearing to dress it up a little. No tie. But unless you're going to a really fancy restaurant, you don't need one.
A lightweight windbreaker jacket, below the hips if possible and in a dark colour, adds a touch of formality while fending off rain or evening chill. You will take it off in the dining room but it suggests you have dressed for an occasion. Or casually toss a sweater over your shoulders.
At any time except the dead heat of summer, a scarf adds European flair. Buy one there as a good souvenir.