Thanks to all who provided invaluable tips here on where to eat in Venice. I was looking for places with great local food, not too fancy/stuffy and places that are good with kids. We had our hotel book all 3 dinner places below ahead of time, and it did seem that reservations were a must. All three places were perfectly friendly about making plain pasta with parmesan cheese for my daughter, and at Ai Promessi Sposi it came slathered in butter and the homemade pasta was incredible! All three places were full of tourists but then we ate early (7:30-8) and we were traveling in July.
1) Il Refolo: I would HIGHLY recommend this spot, especially for families. Kids can run around outside and there's a square nearby with kids playing soccer. It is truly the perfect Venice location -- outdoor candle-lit tables right next to a quiet canal in an out-of-the-way location, so there's a minimum of foot traffic. We ordered bruschetta, spaghetti alla vongole, both very simple and very good. We also ordered a calzone and most tables got pizza -- nothing special but great with kids. I think my family would have liked to have eaten at this place all 3 nights but i insisted we try some other spots. (By the way, PBSF had advised that we take the vaporetto from Arsenale, which was spot-on -- we had no trouble. Then we walked home to the Castello area -- got lost a few times but it was fun.)
2) Osteria Ai Promessi Sposi: Cozy out of the way spot. Menu all in Italian but a very capable server translated the whole thing for us. We started with gamberetti en saor, which came on a plate with bacalao mantecato. It was our first timing having this, and we had a hard time deciphering what was in it, but it was our favorite dish of the night. The "en saor" preparation is tangy, a little sweet, totally addictive-- I wish I'd tried sardines al saor as well. We also ordered the octopus appetizer, delicious too. Then squid ink pasta, fabulous pasta all' amatriciana, followed by a zucchini lasagna that was highly recommended by our server but maybe a little too rich for our tastes.
3) Osteria Alla Frasca: Really delightful, out of the way place on a quiet square, and we had nice interactions with the friendly owner. Appetizers were octopus and something else that we've forgotten now. Then Spaghetti alla frasca which was our favorite dish of the night (basically a simple seafood pasta but with a TON of seafood and very smooth, flavorful sauce), a gnocchi with shrimp, (didn't like this dish too much), then sea bass entree (good, simply prepared -- maybe a tad dry).
Our lunches were rather random, and based on where hunger struck. The first day we got lucky with a cichetti lunch at a bar on piazza Santo Stefano. The second day we wanted cichetti again but couldn't find something to satisfy my picky eater daughter. We ended up getting slices of pizza at Farini, in the area behind the Rialto market. It was very bready but somehow it completely hit the spot. Good toppings. I wouldn't go out of my way for it, but it was fine and very cheap.
Overall, we loved our Venice trip and would advise anyone going there to make reservations ahead of time and do your research! Thanks to all the Chowhounds for helpful advice. This trip provided the impetus for me to get back on Chowhound after a long time away, and it reminded me what a great community of food lovers this is.