I visited Rome for 3 days back in 2012 while visiting a friend who was working in Germany at the time; this was my first time traveling internationally. We stayed in an apartment in the area known as the Historic Center of Rome and didn't really venture outside this area that much, though many of the iconic elements of Rome - the Pantheon, the Colosseum, the Vatican, and many more - are either in this neighborhood or easily within walking distance.
We flew from Nuremberg to Rome on a Swiss airline that was fairly empty and gave us chocolates. Going over the Swiss Alps was quite turbulent - there was a point at which the plane dropped so fast I actually rose up out of my seat. After landing, we took the train into the city center, as the airport is fairly far out and getting a cab would be pretty expensive while the train is fairly cheap. After getting into the city, we grabbed a taxi at the terminal for the final part of the trip to our apartment. The taxi was less like transportation between two points and more like a roller coaster thrill ride. There didn't seem to be any defined lanes, clear pedestrian crossings, or traffic laws of any sort. If your vehicle fit, you could go there and the general rule for intersections was "fortune favors the bold." We turned off the main roads and onto the narrow streets that are common in the historical center. There's no sidewalk or anything like that, so these streets serve both pedestrians and vehicles. When we eventually stopped, I was unsure as to why - we seemed to be just stopping in the middle of the street; but that's because we WERE just stopping in the middle of the street because, as I said, there are no sidewalks so there's really no border between "street" and "building entry."
It's been a while since the visit, so I don't recall too many specifics about where we went and when. I do know we started our first morning walking over to Piazza Navona and then going on to see the Pantheon, but it becomes a bit more hazy after that. I will say that walking around the historic center of Rome is an amazing experience. Narrow streets are crowded even more with outdoor seating of nearby restaurants and every few hundred yards you stumble across some iconic site that you see on travel shows. For me, Rome was a place that just made sense in terms of navigating the city. Even though the city isn't on any sort of grid and the small side streets can move and curve in any direction, walking around the historic center felt natural - like how cities are supposed to feel.
Here's a (non-comprehensive) list of places I went:
Piazza Navona - a square where a lot of people gather with buskers and other interesting spectacles; avoid the restaurants within the Piazza and walk a block or two outside to find some really great places (I recommend Saltimbocca Ristorante for their great mozzarella appetizer or the nearby Quinto Gelateria for some tasty gelato)
Pantheon - a historic building with beautiful paintings; isn't very big so it's easy to get in and out quickly
Spanish Steps - seems to be a hangout location for tourists; spotted some Space Invader street art while walking up the steps
Vatican Museum + Sistine Chapel - the Vatican Museum is a must see, though expect a long line; the place is filled with beautiful sculptures and paintings that cover every available surface (ceilings included); I will say that I did not love the Sistine Chapel experience - there's no talking and no photography, so it's an extremely packed room where you probably won't hear anyone talking but you'll definitely hear someone loudly shushing others
St. Peter's Basilica - the building is beautiful and the square outside is lined with columns topped with statues; if you're heading here from the historic center area, take the St. Angelo Bridge, which is lined with statues of angels
Colosseum - obviously will be packed with other tourists, but it's one of the most iconic places in Rome; the Circus Maximus is nearby so you can see both of these spots together