Here's my suggestions based on our having taken our 8-year-old and my mother to Italy for 3 weeks in July 20 years ago, and our 6-month-old and just-turning-5 grandchildren to Italy last year for 2 weeks in March-April.
In general, do stay longer in fewer places, especially if you will be traveling by train between towns. If you rent a car for Tuscany, it will have to be a van to hold all of you, or (actually cheaper) rent two cars. We had to rent a 9-passenger van for the 6 of us last year.
Our son loved our week in Venice and still remembers it. The canals, the boats, the bridges, being able to walk freely along the "streets" without any danger from vehicles (not even bicycles!), two bell towers, and even the architecture and a few churches. There was no point in trying to take him to a museum or art gallery, so we didn't. (His older brother we would have taken but he was in China at the time.) But he did enjoy much of the art in situ, like Saint George and the dragon. You can pop in and out of such churches and other places instead of having to commit yourselves to a longer time in an art gallery. We stayed in an apartment in a very un-touristed area of town, which I think is an essential thing to do, especially in the hot and crowded summer. Be sure to have air-conditioning --- it is very unlikely that you will be able to open the windows at night. Noise and mosquitoes.
I would add more nights to Venice because the kids will likely be jet-lagged and awake at night and so will you. Our younger son was (and still is) a very picky eater and survived mainly on gelato, bread, cheese, fruit, and pizza --- this was one reason having an apartment with a kitchen was great. Another reason an apartment is better for a family group is that everybody has their own rooms but there's a common space --- you are not trying to sleep in the same room as your kids or having them or the helpful grandparents down the hall in a hotel. When you are looking at any accommodation, think about where the one-year-old will nap. Look carefully at reviews and photos to make sure you actually get the number of bedrooms you need --- a pull-out couch in the living room is often listed as a bedroom. On several other threads, there's whole long discussions about stroller vs backpack in Italy --- just search on "stroller." If you do Airbnb, look for a Superhost who speaks English and is helpful --- we have found these hosts to be way more helpful and fun than anyone working at a hotel.
As for the Tuscany/Siena/Florence part of your trip, an agriturismo is a good idea or just a regular B&B in the country or a self-catering villa in the country --- this is assuming you rent a car or cars. I would probably spend half your time in such a place and the other half in a smaller, less heavily touristed town such as Lucca or Montepulciano. What's nice about staying right in the old part of the a town is that your family group can easily split up in different ways and do different things more spontaneously throughout the day instead of having to decide "Do we all drive to the beach today or not?" or whatever. You will have to park outside the ZTL, but there are often parking lots right outside the old walls.
Our 5-year-old grandson adored Italy and still talks about it. You will not be able to see everything and do everything the way you would on an adult trip, but you will enjoy watching the kids fall in love with another county and another culture. All of your children will be welcomed in restaurants and will be amazed at the pedestrian-only areas inside of towns. And there's gelato. At least once a day.
I would just skip the Cinque Terre in the summer. But something like 750,000 other people disagree.
So, fly into Venice and out of Florence. Take the train from Venice to Florence and then on to Lucca or other town, and rent a car or cars only at that point.
