Two full days in Rome are enough to cover the ancient core, the Renaissance center, and the Vatican without sprinting the whole time. This plan splits the city into an ancient-Rome day and a Vatican-and-piazzas day, so each one has a clear anchor and room to actually enjoy it.
Arrive early at the Colosseum before the tour groups build up, then walk straight into the adjoining Roman Forum and up onto Palatine Hill for a view over the ruins and the Circus Maximus below. Break for lunch nearby, then spend the early afternoon at the Capitoline Museums on the square above the Forum. As the day cools off, head into the center to toss a coin at the Trevi Fountain, step inside the Pantheon, and finish with a slow climb up the Spanish Steps as the evening crowd gathers.
Book a timed entry for the Vatican Museums and Sistine Chapel and go straight in at opening, since this is the one stop on the trip worth building a whole morning around. From the museum exit, walk to St. Peters Basilica and climb the dome if you have the legs for it. In the afternoon, cross back over the river toward the Vittorio Emanuele II Monument, then wander into the smaller churches nearby, including Church of Sant'Ignazio di Loyola and Church of San Luigi dei Francesi, both a short walk from the Pantheon area, before winding down with an evening stroll through the surrounding piazzas.