Disclaimer: I've been to Rome before and visited all of the major sites (Colosseum, Vatican, Pantheon, Forum, Trevi Fountain etc.). Those are all places you should go, but won't be listed here. You already know plenty about those places anyway. ;)
Aventine Hill- A beautiful, peaceful walk up Rome's southernmost hill will bring you to the Key of Malta where you look through the keyhole and see a perfect image of St. Peters Basilica. You feel like it's a secret place only for you.
Protestant Cemetery- The cemetery is the resting place of Percy Shelley and John Keats, and it also holds the Pyramid of Cestius, built about 18-12 BC as a tomb for Gaius Cestius, it is one of the best preserved ancient buildings in Rome, and very interesting to see. "It might make one in love with death, to think that one should be buried in so sweet a place," wrote Shelley, not long before he drowned and was buried here.
Largo di Torre Argentina- This is the location where they believe Julius Caesar was assassinated, and it is now a cat sanctuary. You can see many cats roaming the ruins while thinking about Caesar being stabbed many times and falling to his death at this very location.
Tiber River- Take your time and walk along the Tiber looking towards the beautiful city of Rome, and it's especially nice to cross the ancient bridges that still exist.
La Carbonara- Try as many pastas as you can here! They are so fresh and delicious. I had cacio e pepe and aglio e olio, and both were so yummy! I highly recommend those dishes. The atmosphere is cozy with graffiti on the walls, and the service is great.
Hostaria Romana- This is the place you go for traditional meals. I read that they have great Trippa alla Romana, so this is where I went. The dish was cooked perfectly, and the service was attentive.
Gelateria del Teatro- When in Rome you must always eat gelato, and this place is fantastic. They had me at dark chocolate and red wine gelato! Why doesn't every place do that? It is the perfect combination, and it tasted so fresh and wonderful. You have to try it!
Baylon Café- This is a great place to stop and take a break in the middle of running around and seeing everything you want to see. They have a delightful meat and cheese plate that is the perfect snack to give you fuel to keep on going. They play good music and the service is friendly.
Mecenate Palace- This is a hotel that must have been very fancy back in the day. It's a little tattered now, but they offer great rates, it's comfortable, clean, quiet and the location is ideal. It also has a fun rooftop bar where you can look down at a square with Basilica di Santa Maria Maggiore while you sip on some Fernet Branca.