If you're spending a few days in Rome then you should plan to spend one of those days at the Vatican. Vatican City is its own independent city-state. It prints its own stamps, mints its own euros, issues passports, and has its own anthem and flag.
While visiting the Vatican be sure to check out St. Peter's Basilica, Michelangelo's Sistine Chapel, and the Vatican Museum. If you're in the area on a Wednesday, then consider attending the Papal Audience. This gives you an opportunity to see the Pope. The Audience starts at 10am but people start arriving around 8am to get the best spots.
Vatican City has been added as a destination in the Duckwyn passport app.