May or September

What part of Italy? There is very different weather from the Dolomites south to Sicily. Do not worry about it being warm enough in May. But then, I would not find the low 80s very comfortable, especially if humid.

We are thinking four nights in Florence, three nights on Capri and three nights in Positano.

while we have not been to europe in september, we find May a great option. the daily temps are nice ( a bit fresh at times) and we have swum in the sea ( again, last year in positano- just a bit cool) having driven and stayed in various parts of italy etc. The accomodation is modestly priced. ( probably less so the later you go into May)
the advantage of may is everything is green and 'fresh' with not too may crowds ( although I think lots of people are also thinking the same) the snow is still around in some places and the rivers have snow melt etc. driving around is good ( no snow tyres etc)
Early May would be even better, possibly less cost and people. expect lots of people on the 'coast' in late May.

look at what is happening in europe now. Wow 46C in spain. all drying out and too hot.
we have experienced some unexpected 'warm' days even in May.
IMO, with your first trip , go and enjoy the spring freshness.

hope this helps.

Early September is likely hotter than the low 80s. Late May is more likely to have the temperatures you desire. However no one can tell this far in advance.

We were in Italy in late May 2014. The highs were in the low 70s every day. We returned in late Sept. - early Oct. last year and the highs were in the upper 70s every day.

I'd spend some quality time on wunderground.com, choosing History > Monthly to access actual day-by-day temperatures for each of your destinations. There will be at least 10 years' worth of data. This will be a lot more meaningful than averages or individual people's experiences. Published averages on Wikipedia, just as one example, seem to be based on data no more recent than 2010. Guess what has happened to temperatures since then. If weather research (including rainfall data) still leaves you undecided, I would choose May for its considerably longer days.

I could take either happily. Both should be rather warm during the day, but in late May, you have cooler nights and even cooler sea temps.

The water will definitely be warmer in early September.
Late May would be my preference because of the longer daylight, but it’s a toss.