If your kid is a picky eater, traveling to India or Japan might be a bit tricky. My boys are sort of average eaters and Bijou has broadened her palette and is now what I consider to be "average." With that said, dining in Italy will satisfy any carb-loving kid or for those on the all-white diet.
Our favorite meal of the whole trip was at a restaurant called La Zucca. The menu was sophisticated enough to satisfy the adults and had staples like spaghetti to satisfy the kids. My "rules" on the trip included the obvious... no screaming in the apartment or standing on the edge of the canal (neither of which was followed 100% of the time) and to eat gelato and pasta daily (which everyone had no problem following 100% of the time). The kids also enjoyed their fair share of Italian hot chocolates called, Cioccolata Calda, which were so thick they had to be consumed with a spoon.
My tips for dining in Venice is to avoid restaurants that offer you a menu in English and places that have photos of the food in the window. The later is obvious I'm sure, since the same rule is true in the US. English is widely spoken in Venice, especially by those working in restaurants, and the waitstaff is happy to translate their Italian menu if you are stumped. I personally enjoy the opportunity to order a "mystery meal" but my kids feel quite different.
All in all, it was a refreshing change dining with kids in Italy than in France. In France, dining out with your young kids is unusual and in some instances not tolerated. I've experienced several acts of rudeness that I just have to laugh about now. One time my boys were kneeling on a bench seat at a cafe in France so they were tall enough to reach the table. The waitress didn't like them having their shoes on the bench so she came over and scolded them and told me that they needed to keep their feet off the bench. Another time, a waitress nearly barred us from entering her restaurant when our posse came toddling up to her front door. We made our way in anyway and proceeded to have a well-behaved meal which won over the likes of our child-hating waitress and by the end of it she was blowing kisses to E + W. Oh, and I've lost count of the amount of times my family has walked into a restaurant only to be told they were "ferme" even if the sign read "ouvert" or an empty restaurant where we were told that every table was booked. Very suspicious! Needless to say, dining in Italy with young kids is accepted and accommodated, making it less stressful and enjoyable for the adults.