This one is mentioned in Rick's Rome book, but it was fab. South of the Pantheon. Trattoria de Pallero. Sit down and let Grandma and Grandpa feed you, no menu. 25E each and we sat for several hours and enjoyed amazing food and wonderful hosts. There's another one similar called Alfredo E Ada that we did not get to, but that gets excellent reviews online.
In the CT, Rick mentions the Pirates, but not how fantastic their dinners are. Up the hill from the train station. We enjoyed 2 breakfasts and one dinner with them. The pannacotta and canoli are not to be missed. If you can, have the thick hot chocolate. We couldn't fit it in--too stuffed, but we watched a couple of students have canoli and hot chocolate for dinner and were envious.
In Venice, we just wandered,and ate where we saw a menu without "tourist" and 12 languages on it. One day for lunch we ate with all the city laborers and hospital workers at a place called Birreria Barbanera. Yummy. If you can, hit the cichetti bars and eat and wander.
Avoid Rick's recommendation on Murano. Crappy food with a high charge and service fee! I wish we would have followed our instincts and gone into the local joint.
Have fun!