What we saw in Rome, was simply astonishing. There's absolutely no way to avoid all of these great places - even by taking a different way you'll see something really fantastic en route. To me, even though Rome doesn't have such great museums as London or Paris, just walking the streets does the trick (if one doesn't mind the traffic).
I think we spent a good part of our stay walking through the narrow, underlit Roman streets - without a plan in the evenings, just wandering from one piazza to another.
The Roman piazzas have lots of magnetism to them - San Pietro, Navona, Spagna or Campo De Fiori, to name only a few. Mostly, they're quite crowded all day long, but then, the Campidoglio, being one of the most beautiful (no wonder, that's Michelangelo's doing) is almost deserted. The problem is most of these just urge you to sit down with a coffee or a glass of wine and relax - when there's still so much to see..