My Travel Notes have a small blurb about public pools in Florence. They typically open when schools get out in early June. My favorite is located in the Cascine.
I have heard that a water park is opening near Florence, in Campo Bisenzio or Sesto, I think. I figured I'd hear all about it when I arrive next month.
I don't know anything about public pools in Rome. I have taken my nieces to Hydromania, a water park on the outskirts easily accessible by metro and a bus connection. Here is thread about that.
The pool at the cascine or at Bellariva would be your best bet. The waterpark outside Florence is called Hydron. It is the largest in Europe, so they say. I have looked for it on the web several times and found nothing but I know it opened.