After you've been issued a permesso di soggiorno, you must apply for residency (residenza) within 20 days. Visit your municipio (city hall) and find the anagrafe office. They will need your passport and permesso di soggiorno - some towns may request a rental contract. Your information will be entered into their computer, the application will be printed out, and you must sign.
Next comes the wait for the police to visit your address. Sometime in the coming weeks, they'll make an unannounced visit to confirm you actually live at the address. A few days after they visit, pop by your anagrafe and see if your residenza is complete. If so, they will print you a certificato di residenza (useful for many things - signing
up for health care, getting a drivers license, reduced utility rates, etc). You may also apply for a carta d'identità, which is a handy photo ID with your address, etc.
Please note: residenza anagrafica (as described here) is a different animal than tax residency. Tax residency is beyond the scope of this article - please consult a tax expert or commercialista for details specific to your circumstances.