You can go to trenitalia.com and check on current rail fares, which will allow you to figure out that segment of expenses fairly accurately.
Except for Florence, the places you are going aren't heavy on museums, so entry fees probably won't be a huge expense. If you know what Florentine places you want to go inside, you can check their websites for entry fees; you might or might not save money on a museum pass. Bolzano has the South Tyrol Museum of Archaeology, a really excellent museum housing the Iceman, which I'm almost certain would interest the adults and your teenager. The current price of a family ticket for all of you is 18 euros.
As of 2015, you would have been safe with an average cost of 2 euros for a snack-size serving of carry-out gelato; I'm sure I paid a bit less. It could be somewhat higher now. The main thing is that it will cost you much, much more if you eat your gelato sitting down in a café. The same goes for coffee, cold drinks, etc. Unless you seriously need to get off your feet, buy your refreshments at a bar or a carry-out stand.
Restaurants in Italy are not dirt-cheap. Italians care about quality and are willing to pay for local ingredients, etc. You'll see menus posted just about everywhere, so you can judge how costly it might be before going inside. Pizzerias will usually be about your least expensive hot-food option. You can run up quite a tab on bottled water and sodas at any seated establishment, so be careful there. What I try (and fail) to remember to do is drink a lot of the water I'm carrying around with me before I go inside a restaurant so I don't walk in dying of thirst. That way, I hope to get by with just one bottle of water rather than two. It will be very hot in July; you'll need water, but a restaurant is the most expensive place to buy it. In most cases a little corner store will be cheapest, though not necessarily if it's across the street from the Vatican. Next best will be buying and consuming the cold beverage at a bar. A few years ago that could cost as little as 1 euro in non-touristy neighborhoods.
You may find tasty snacks at street markets. You may also bump into shops selling prepared foods to consume at home. I don't remember what they're called, but it will be clear what they are from the window displays. A lot of the food is sold by weight, so if you don't go overboard you can put together a nice, moderately-priced picnic. You might need to eat the food in your hotel room (if that isn't forbidden), depending on the local customs. Some cities are fining people who plop themselves down in public areas to eat.
