Start on an off-on bus. It will take you by all the major areas.
Of all tours, I think one of the forum is necessary. It just will make little sense without one.
For such a short time I'd try and not cram everything in.
The lines at the Vatican are not most any line you have experienced. You could go with a tour group there before you left, but again I think you would be better letting them choose.
I once took my 3 teenage daughters to Rome.
They wanted "to see" a lot, but not necessarily tour most places.
Piazza navona (day and night), Trevi/Spanish Steps, and having their picture taken outside the colloseum were their highlights.
That is,except for shopping. They loved the crowds and the fact that I would leave them alone in the Navona area for a certain amount of time. Ditching their parents was also a popular thing for them to do during our trip to Europe.