Ciao Robin and welcome to Florence as a resident!! Glad you found us, although I have to be honest and say that the majority of our visitors and members are "passing through" much shorter stays and thus our specialty has become info on short-term visits.
Long term accommodation is definitely a bit harder to come by, but not impossible. The difficulty lies mostly in the types of contracts that are available for long term - many being years long.
For a year stay, you kind of fall in the middle - most private renters won't rent for that long, so your option will almost always be to deal with an agency. And while there might be an extra cost in doing so, I highly recommend it - it is much preferable to deal with an English-speaking person for any needs you might have with the apartment, especially when there are troubles such as furnace acting up or water pipe broken!
So to start your search, I would recommend you get in touch with Riccardo Gogoli at MerryGoRound - while they might seem at first glance to cater to short term visitors, they are an agency that deals solely with apartments and also handles them for local foreign students who are here for less than a year - so the right type of agency for you. This is their site:
http://www.merrygoround.org/
Another agency that also deals with longer-term rentals is Pitcher & Flaccomio. There rentals are more higher end and they don't have all of them on their site -
http://pitcherflaccomio.com/ - so the best course of action is to contact them directly. I would recommend talking to Corso Flaccomio at sales AT pitcherflaccomio.com, he's in charge of the real estate part but am sure will help pass you on to the right person (but I know him ;-) )...
As far as schools go: are you sure you want to put your daughters in local public schools? All of the expat families I know (I only have a 3 year old so still learning how the system works as she grows) who are here for 2-3 year contracts have placed their kids at the American International School in Bagno a Ripoli (even non-American families). It isn't a cheap option since it is a private school, but in that way they have some continuity in their studies while you're here. Italian schools will do their best to welcome your kids, but I assure you all teaching will be in Italian. So they'll have the double challenge of learning math, science, etc in another language. Another possibility is homeschooling for that year, but that would mean a lot of work and preparation for you.
I belong to a great Moms network here in Florence composed of international expat moms with English as the common language. They are a wonderful resource, so I would highly recommend joining in to the community, then also asking to be added on the Closed discussion group on Facebook where you can also post questions. It is likely someone else with bigger kids will have some more useful suggestions than me as regards schools. The network is all volunteer, even the blog, run by Kimberly (tell her I recommended the site) but you can see there is a good amount of basic info already on there:
https://firenzemoms4moms.wordpress.com/
Hit the "Join Us" button at the top to request to join the group (it is like this so that we can share our private details in a secure place - once you're looking at where and when, etc, it is better to not post on public forums ;-) )
Hope I've helped you with some starting places to look for accommodation and that things start to come together! Moving here is a big life event, very exciting!! See you back on here or on the FB group
