Summer in Italy can be unbearably warm or more tolerable, according to where you are going.
Northern regions are usually less warm than the Southern ones, this is the general idea. Yet there are a few exceptions. Make sure you are aware that even if you spend only one day in Milan it might be terribly hot and humid. My sister usually states that Milan isn’t less warm than Naples: it’s only way more humid.
If you want to go to the Navigli area to experience Milan nightlife (please don’t forget the Italian tipping etiquette), make sure you always bring a good bug spray and carry it at all time! Mosquitoes are a huge problem in Milan in summer… especially at Navigli!
Yet not all the Northern regions in Italy are this warm in summer. The Dolomites, Alps and Apennine mountains are always going to be fresh, even in August. Make sure you bring a light jacket or a stylish trench with you that you’ll wear at night in case it gets colder. Of course it won’t be snowy, but expect some rain!
We have had a few years of hard rain in summer on the Dolomites, then a couple of years of 30°C and no rain for weeks. Unfortunately the weather forecast is more and more unreliable!
A rule of thumbs would be to just bring a light jacket and maybe a foldable raincoat that won’t take too much space in your luggage. As a summer mountain vacation, I’m assuming you will be hiking a lot: in this case bring with you a good pair of hiking boots as a regular pair of walking shoes won’t be enough. I prefer boots to shoes because they make me feel safer, as they hug my ankles and protect me from ticks (which terrify me!).
In a nutshell: usually the Northern regions are warm, unless you’re planning a summer vacation in Italy exploring our mountains. Yet you will possibly face lots of muggy days in the big cities.