You have to apply for and timely renew a residence permit. Documents which are required depend on the type of the permit. For work permits you need to demonstrate that you income is above the social minimum. The permit gives you the right to stay on the Italian soil and travel in the EU.
You need to rent a place to live and apply for the residency. Strictly speaking, residency is optional, but it will give you an Italian ID card, and will be required to receive a driver's license. IIRC you have six months before you need to get a local driver's license. Residency will also reduce utility bills and is required in some other situations.
After X years of non-interrupted residency (where X depends on your country of origin) you can apply for a citizenship. And it's a whole new story.