I'll have to dig up my old notes on where we went. For sure, get off the touristy squares and into the back streets. There are some real gems.
We did have a hot chocolate (ciocolotto?) on St. Mark's Square - could have been at the Café Florian - 12 Euros but worth every drop. It was a pot of melted chocolate and a pot of foamed milk, and you mixed as you wished. It easily serves two.