The Centro Storico is what people mean when they say Rome, and the trick to it is to walk the lanes between the famous things rather than bolt from one to the next. Two and a half kilometres, best in the early morning before the squares fill and the heat lands.
Start in Piazza della Rotonda with the Pantheon's portico in front of you — go in early, stand under the oculus, and give it the time it deserves before the crowds arrive. From there the lanes do the work: west to Borromini's spiralling lantern at Sant'Ivo, then into Piazza Navona, laid out on the oval of an ancient stadium, with Bernini's four-rivers fountain holding the middle. You don't have to stop for every fountain, but stop for that one.
North to Sant'Agostino for the Caravaggio, then the small wonder to end on: Sant'Ignazio, where a flat ceiling becomes a dome if you stand on exactly the right spot. From there it's a short walk to the Metro at Spagna and the line home.