I am Roman and I love my city, although it is stressful to live in Rome.
Turists flock to the Vatican or the Colosseo etc
The Rome I love most is the museum in open air, the Rome of Roman Palazzi, and Villa Borghese.
If you stroll in the trident (Babuino Corso Ripetta which start from piazza del Popolo) and get lost in the maze of ancient narrow streets of the center, the urban landscape you see is the same that was, say, in the eyes of Caravaggio about 4 centuries ago.
There are about 400 hundreds ancient palazzi in Rome, many of them are indeed royal palaces. Some can be visited. One example is Palazzo Colonna where the final scene of Vacanze Romane was filmed.
Then there is Villa Borghese and the Museo Borghese (very crowded must be booked). Many consider them the most beautiful urban park and musem in the world. Unfortunately Villa Borghese is not taken case of as it deserves by the City.
Sorry I can't post photos because my PC is broken. I hope to recover it pretty soon.
BTW I have seen many excellent images in this thread. Thank you so much and my compliments.