Which month is best

The last week of May if you want to include a visit to the Dolomites.

It so depends on exactly where you want to go and what interests you. Do you have a sketched out itinerary yet?

In general, I'm partial to months with more daylight hours, so between your two choices I'd prefer May. However, my trips to Europe usually extend into September, and it's only toward the end of the month that I become very aware of the shortening of the days.

If swimming's a big thing for you, check water temperatures; I think they'd be considerably warmer in September.

The Wikipedia entries for most cities have a climate-summary chart. I find those useful for comparing average rainfall in different months.

Edited to add: In southern Italy it can be quite hot in both those months, especially in late May and early September. Folks heading south who don't like the heat might well prefer April or October.

For us, it would be early May. But we aren't interested in the Dolomites or swimming. Or very late September into early October.

It depends on where you want to go. I like May because of spring is in full bloom, temps are warm but not yet hot, and daylight is longer. If you are going to the sea, water temperatures are kind of cooll still.

All of our trips have been May./June. Depends upon where you are going. We stayed in a beautiful agritoutousismo in Alba and it was wonderful. Another trip two weeks later it was hot and there was no air conditioning.

Do your research.

If your going to the Alps: first part of September has less chance of rain than May. J

This is a great question that you pose Les. Worthy of a serious travel site's forum such as this.
Lots of great input for you above.
I am done. the answers

I believe late September/early October first, then the first 2 weeks of May.

My vote is for May--it is sunny longer and not as hot.

I travel to Italy in late April or early May. Although I live in L.A. where it is very warm in the summertime--I do not do well in humid weather. It is humid in Italy.

My first trip to Italy was in late May through early June and thought I was going to die during with the heat/humidity.

Depends on where you are going.
We have traveled to Italy in both months and here is my take:
If you're visiting the Amalfi Coast/Sorrento then May/June, but these are very long days and with the sun setting late we are out and about very late. Very much slower travel due to the heat with more Spritz/beer breaks during the day. We were in Italy in June and man was it HOT, my energy was zapped and I felt like a hot sweaty mess at the end of the day. BUT I still loved it all, would probably not do Rome in that month again, but would stick to the coastal towns.

If we are just traveling in general to non-beach spots, then September/October, this means we are out the door early morning and back in around 8:00PM a very different travel style than in the Summer. Not many breaks knowing our day is ending early, but more comfortable due to the weather.

If I was to do a mix of both I would probably choose late August/early September. Doing the coastal town first then the interior.