What do people recommend for clothes to include for early September in both northern and southern Italy?
For me it would be summer clothes and a light jacket or pull over as needed.
If you look up every city on https://www.wunderground.com/ or some other weather site it will have the historical averages.
Then pack according to that as everyone here has a different barometer of what is hot, warm, cold etc.
Expect rain - Even you luggage can be soaked when the sky over Italy opens up.