Hi,
congratulations on your purchase.
I don't think there's an average price, well advertised commercial sites tend to be dearer than others.
We have three kids, and are members of the Camping and Caravaning Club. We tend to use a variety of sites, ranging from £5 for a certificated site, which can be a field with a manhole cover and a tap, to £24 for a commercial site, with pool, playarea etc.
We tend not to use club sites, as we never found them child friendly particularly, but especially as we tour, and usually phone on ahead to book a single night, and extend the stay if we like it.
The same applies in France, we gone from a few Euros for a farm campsite in summer, to £35 per night in high season, at excellent sites, prebooked, by the beach. (We have to, they're usually full in high season)
Can't help with Italy I'm afraid
Mike