I’m going to renew my Permesso di soggiorno this April and I wanted it to be valid for traveling around Europe, for now I have Permesso di Soggiorno (Attesa Occupazione).
I assume since you have a residence permit that you are a non-EU citizen.
From the Europa website on "Travel documents for non-EU nationals", it says:
Travel documents for non-EU nationals
Passport and visa requirements
If you are a non-EU national wishing to visit or travel within the EU, you will need a passport:
- valid for at least 3 months after the date you intend to leave the EU country you are visiting,
- which was issued within the previous 10 years,
and possibly a visa. You should apply for a visa from the consulate or embassy of the country you are visiting. If your visa is from a Schengen area country, it automatically allows you to travel to the other Schengen countries as well. If you have a valid residence permit from one of those Schengen countries, it is equivalent to a visa. You may need a national visa to visit non-Schengen countries.
So since you have a residence permit from Italy (Schengen country), it should be valid to travel to other Schengen countries so long as you also carry your passport.
Note that, as mentioned above, not all European countries are a member of the Schengen area, so visa requirements would be different for non-Schengen countries, (but since you've tagged the question as 'schengen' I presume you know this :) )