I believe that with a Eurail pass (which is available in short-duration format and is valid for first and second class coaches), you get access to the TGV. So you could probably make this trip in one day. Rome to Turin and on to Nice (or wherever the last TGV station is in France).

A copy of Cook's European timetable would be helpful. (I only know of one store where it is readily available - Forsyth Travel near Kansas City - but there must be other travel shops with this very handy guide.