These pop up like mushrooms and we don’t have any one especially to recommend. Most have brand-spanking-new computers, long hours, and friendly service; others have the opposite, where you’re lucky to get an @ symbol on the keyboard. With the new laws against terrorism, everyone must get a card (tessera or scheda) before using the Internet, and to get one you need your passport or some other form of photo id.
Free wifi is also available in the piazza outside the Informagiovani office (see Informagiovani Perugia), and in Piazza Italia and Piazza 4 Novembre thanks to Perugia city (The wifi network is called O75, and you need to follow the registration to obtain user and password)