While there are many local specialties that everyone will recommend, like orecchiette in Apulia, I want to mention some that do not get mentioned as often.
Egg soup, Salina: A soup with poached eggs and vegetables.
Fried cheese, Abruzzo (and other areas): This fried cheese is a bit like haloumi in Greece, but not as salty. Served melted from the oven. In Salina, it is deep fried.
Pizzeli (wine cookies): Most regions of Italy have round hard not sweet crackers that are for dipping in wine.
Egg cheese balls, Abruzzo: These were like a cheese egg souffle in a round donut hole size and shape. Served with red tomato sauce.
Raw artichoke salad: Actually, try any of the salads and vegetables that you have never tried before like raw fennel or artichoke.
Fried egg tart, Campania: A frittata which is then deep fried.
Taralla (Napoli): These are another baked bread twist. Many places in Italy make them like a small ring but in Naples, they are longer.
Fish balls, Salina: On this island off of Sicily, one can try tender fish balls. Try them. They are surprisingly good.
Pizzette (small pizzas), Rome: Many bakeries sell mini pizzas (many with no cheese or red sauce as a pizza just means a type of bread). These pizzette are often served at happy hour.
Pane Cunzatu, Salina: This is like a giant garlic bread. The plate-sized bread is toasted and smothered in toppings. Absolutely awesome!
The melon-cucumber of Apulia: I wrote about this before. When in season, it is served as the “vegetable” but only to locals so make sure to ask!