Well, I think you are going to have decide between northern or southern Italy? Or perhaps both?
The south is much better if your looking to relax on a beach as the seas are much cleaner. There is lots to see in the south, such as
Rome,
Amalfi coast, etc... You could even visit the island of Sicily.
If you choose the north, I guess you will arrive in
Milan, there is plenty to do and all fairly close by.
I would suggest staying in an area such as
Lake Garda. The lake has amusement parks such as Gardaland, Movieworld and Waterparks as well as other activities on the lake for your son.
If your daughter loves shopping then
Milan is the fashion Capital of Italy.
Other cities that you can visit on a day trip by train are:
Verona and
Venice
In early September, you should still find nice weather in all of Italy.