What are the most interesting places to visit while in Italy?

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Next week I'm going to Italy for a few days and I want to know what is there to visit. I heard of the main places and I will go there too, but other than Vatican, Colloseum, Pantheon, Trevi Fountain, Barberini Palace, Trastevere... what else is there to visit? Also, I need a confirmation that all these important places are closed for visits on Mondays.