A friend and I recently took advantage of one of these free days to see .... a not-free museum. It's a long story, but we did miss most of the crowds!
Now I can finally say that I've made it to the Capitoline museum - apparently the first museum in the world, though I'm sure that title is disputed. In any case, it didn't disappoint, from the courtyard designed by Michelangelo to a ceramic collection that my friend who came me with just adored. Because this museum is split into two large buildings I would suggest that you give it 2+ hours and have lunch halfway through. I'd also suggest picking a few things you really want to see and spending your time accordingly rather than trying to see it all.
In no particular order, here are some of the highlights:
Perhaps even more famous, Romulus and Remus being raised by a wolf. This bronze has become a symbol of the city of Rome
A bust of Medusa by Bernini
A somewhat graphic depiction of a lion and his lunch. (The theme of animals brutally killing other animals came up a lot in this museum for some reason.)
Finally, there were a weirdly large number of paintings featuring St. Sebastian. Not sure who's choice that was. I really wish someone would do a ranking of them all, in order of actual martyrdom, because he usually looks like he doesn't particularly mind being shot full of arrows.