In Rome I would split up your time and see the sites that are close to each other on the same day.  For example,

  1. Vatican Museums, St. Peter’s, and Trastevere, (artist quarter) seeing the church of Santa Maria in Trastevere for its incredible mosaics
  2. 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
  3. Piazza Navona, Pantheon, and then walk to the Piazza di Spagna (Spanish Steps), another walk to Pincio (park) with the Villa Borghese art museum.