Al Covo in Venice is great, owned by a lovely couple (one American, the other Italian)... seafood, romantic, great food. Try to eat in the garden. Also Corte Sconta is very good, hidden and a bit rustic but excellent food. There are lots of bars that serve wine and apps, like Al Marca, Al Prosecco, and Do Mori. We like to alternate; we do a fancy lunch one day and wine bar in the evening, then we'll do a picnic lunch the next day and a dinner at a nicer restaurant that night. For your picnics, I highly recommend saving a couple of 1-1.5 liter water bottles and hitting one of the wine stores that sell wine out of the cask. Great and very cheap, it's what the locals drink.
As for Florence, there's a place which you CANNOT miss: Teatro del Sale. It's a private club which you join for a few euros, then it's a big buffet with fantastic food and also different types of performances are offered. You sit with others, usually locals, which is nice especially if you speak a little Italian and/or they speak a little English.
Florence has changed a lot over the years, it's not what it was... lots of high end chain stores and unfortunately most of the small artisans have left town. I definitely suggest renting a car or taking some kind of tour down to Chianti. It's hard to find a bad meal there.
Buon viaggio!
Lynn