Venice

I felt like this was the hardest city to take a toddler to. She wasn't fascinated, like I was, by the fact that police, ambulances, taxi's, appliance delivery, and funeral hearses are all done with motor boats. She didn't find the glittering ceiling at San Marco's to be breathtaking. She did stop to gaze at the quaint bridges spanning narrow canals that weave through the city or smile at the dead-end streets that lead straight to the water. Instead, she was restless in a very crowded, very tight, but very beautiful place.

Don't bring a stroller
Venice is a city of foot traffic only. There aren't cars, which is nice, but it also means the streets are very narrow and full of stairs. You cross bridges every few minutes as you walk down the island. (It's truly incredible, just not the most stroller friendly environment). We opted for a baby backpack (my dad snagged one on KSL for $2) and it worked great for us. Ada slept a bit and liked being up with all the adults. We were in Venice with a stroller in March. It was bearable (and I did appreciate having a place to change Ada by lying the stroller flat and putting down the travel pad), but especially if you're going in the spring time/tourist season, I think it could be a serious hindrance.

Get a 12-hour transportation pass and a good map
We made the mistake of only getting a one-way ticket for the vaporetto (the water-buses which are Venice's form of public transportation). If I could do it again, I think the day pass would be worth it. The most amazing part of the city was being on the water. The island itself, as I mentioned is weave-y, narrow, and not easy to navigate. We tried relying on the few maps from my guide book which were sufficient, but we did get lost a few times and wished we had something better to help us find our way around. Getting lost isn't fun for parents or for toddlers as it means more time in the stroller and grumpier adults to hang with. Ada also seemed to enjoy the wind in her face and the splashy sounds of the water.

Prepare to entertain, mark some small campos on your map
Because it's not the most toddler friendly and most places are too crowded to just let Ada roam, we did a lot of entertaining. Songs, snacks, little toys. When we did let her down, it was when we found a quiet campo (the Venetians term for piazza) where she could stretch her legs a bit.

Toddler-friendly places in Venice

The vaporetto
Both Mike and I loved the vaporetto. It really gives you a chance to see how unique the Venetian water culture is. It also seemed fun for Ada. The sounds, the motion . . .
 
Piazza San Marco
Though it is very crowded, it's full of people feeding pigeons and Ada got an absolute kick out of all the flapping commotion. (Though my Italian teacher last semester did warn me, very seriously, that pigeons have parasites . . . all pigeons??) If you're good at sticking to your kid like glue, this is a fun place to burn some energy.

Murano
We didn't get out to the island (refer to vaporetto pass section) but I have heard that Murano is the most kid-friend island. This island is where the famous Venetian glass is made. Maybe watching glass blowers could entertain. There are also a few public parks on the island. It mostly seemed appealing to me because it would be less crowded than the main island of Venice.

Santa Maria Gloriosa dei Frari
This was one of my favorite places we visited. The church was quiet, not too many people, and the surrounding streets were relatively empty. Maybe my best advice when traveling with a toddler: find places to take a break from the crowds.

Here are a few other  I read up on:
If your kids are a bit older . . .
Forum on Trip Advisor on toddlers and Venice
Lots of people recommended buying a Venice WC card. Some bathrooms have changing tables.
A few parks mentioned:  Parco Savorgnàn (Cannaregio, near the station and Guglie bridge), Giardini della Biennale, and Giardini di Sant'Elena, and a couple on Murano