Those are all sunny and warm places in Junes, but temperatures will probably swing a lot as in the last years weather in Italy (well, in all Europe actually) has been quite crazy.
Total temperature range in your trip will be from 16-17 degrees at late evening/night to a maximum of about 30 degrees in the hottest moment of the day. Recommended is an umbrella, and a k-way or equivalent: rain can hit very hard and very quickly.
If I were you, anyway, I'd avoid the short pants. You say you are from South East Asia (Philippines, right?), so you are used to hot climate and high humidity. I've been all around Indonesia, and noticed that short pants do not seems "mandatory" between people even if it's really hot. I've friends, too, living in Italy and coming from South East Asia and I've never seen them wear short pants in summer, so it generally seems that short pants is not that much a need. But reason I'm pointing at this is that by wearing short pants you will be denied access to many churches you may wanna visit. The tank top will be a problem, too, but that you can always cover with the k-way or a large foulard (it is quite common)
So, given that:
- short pants could be "problematic" from a touristic point of view
- at late noon you'll probably feel a bit of cold with them
- in June it is expected to have unexpected storms
- it is not like 90% of people and tourists you see in Italy wear short pants, quite the contrary
- one does not simply wear short pants without a matching pair of sandals and socks [cit.]
I'd say: tank top, long light pants, comfy shoes, a foulard, an hat, a foldable umbrella, a k way, and you are good to go.