Rome has a rich history pertinent to western civilization. There are three different aspects of roaming in Rome, namely, architecture, history, and religion (Catholic - Christians). I was profoundly interested in history and to some extent, in architecture, and that influenced my itinerary.
Day 1 - walks
Start with a walking tour and after walking tour use Rick Steve’s free audio tour to wander in the heart of Rome, primarily, Trevi Fountain, Pantheon, and Spanish steps. Then use Rick Steve’s Jewish Ghettos walk to do a self-guided tour of Jewish ghettos.
Day 2 - History
Wake up early morning and head to The Colosseum; it opens up at 8:30, and you better queue up by 8. Do either a paid guided tour here or use Rick Steve’s free audio tours. If you are interested in history, spend less time at Colosseum and more at The Forum, which is rich in history. If its a weekend, you can head to Domus Aurea and visit that too.
Next door is one of the three churches which lays claims to having St. Valentine’s skull, free to visit.
There is the " Mouth of Truth" there as well.
Now head to “Pyramid of Caius Cestius”. The only pyramid of Rome. The best views are from the outside, so don’t bother going inside.
Day 3 - Vatican and Trastevere
If you are interested in the Catholic religion, reserve a full day for this and go to Sistine Chapel as well. But for others, a visit to the Basilica of St. Peter, the first Pope for the Catholics. A few hours are sufficient here.
Then head to Trastevere, an area famous for its ton of traditional shops and eateries. Try vegan Cibatta here.
From here, head to Capuchin Crypt, which consists of various structures made out of human bones.