Hi Lostinmedici! I live about 30 minutes outside Florence and am pregnant but don't have kids yet, so not sure how helpful my advice will be, but I'll try since no-one has responded yet!
I think a lot depends on whether you are planning a short or long-term stay, can speak (or are intending to learn) Italian. Florence has a large expat/international community (especially Americans) and especially in the centre you can find English-speaking doctors, lawyers etc if you need them, as well as social clubs and events for English speakers. Generally though, people don't speak English as well as in northern European countries so learning the language is a must if you want to integrate and meet people outside the expat community. In my experience Italians really appreciate it if you make an effort with the language and will help you a lot, even if you make loads of mistakes.
I've been here 5 years and haven't suffered that badly from culture shock, but I'm married to an Italian and can (now) speak the language, which definitely help. By far the most frustrating thing for me is still the bureaucracy. Things like going to council offices, registering for a doctor, and even just going to the post office are all massively frustrating and time-consuming and you get told lots of contradictory information. It depends where else you've lived though, you might be used to that already - I lived in France where they also like paperwork and seemingly random regulations, but Italy is worse! A lot of patience is required, especially when you first arrive.
Generally though I'm very happy here, it's a really beautiful part of the country, lots to do in terms of culture and nature for kids and adults, and and a more relaxed pace of life in general. Italians love children and you can take them everywhere. I've found the state healthcare to be really good so far (had a few long-term things diagnosed here which my UK GP would never refer me for), but can't speak for education. I guess it depends whether you want the kids to go to state nursery/school or are thinking of sending them to an international school.
You could try contacting this expat mum's group who could advise you on schools and similar: firenzemoms4moms.com/