In Rome I would split up your time and see the sites that are close to each other on the same day. For example,
- Vatican Museums, St. Peter’s, and Trastevere, (artist quarter) seeing the church of Santa Maria in Trastevere for its incredible mosaics
- Outside of Vitttorio Emanuele monument at one end of the Piazza Venezia (often called the wedding cake monument or the typewriter), Forum, Trajan’s Forum, Coliseum, and then a walk to the Campo de’ Fiori (flower market). There is a great bakery in the square called Il Fornaio at Via dei Baullari #7
- Piazza Navona, Pantheon, and then walk to the Piazza di Spagna (Spanish Steps), another walk to Pincio (park) with the Villa Borghese art museum.