The most important are "la Repubblica" and "il Corriere della Sera",
www.repubblica.it and
www.corriere.it,every day around two p.m. on the websites are published the articles that were published in the publication of the morning that you can buy on streets.
For economic arguments the best is "il Sole 24ore",
www.ilSole24ore.com.
These for national arguments, but for local issues it is better to read local newspapers like "il Messaggero" for Roma. Every city or region have a lot of local newspaper.