There are dozens, indeed hundreds, of worthwhile places in Italy which could occupy your time for years. You can't reasonably expect to do justice to the whole country in twelve days.
Twelve days is longer than most people spend in Rome, and unless you specifically want to concentrate on that city (which you don't), then I would consider a two-centre holiday. This will give you some contrast. Typical advice on this forum for for a first-time visitor is four full days for Rome (I mean full sightseeing days), to which you can add day trips for Tivoli and Ostia Antica if they interest you.
This leaves you enough time to move on to Florence. There's enough to do there for at least three full days, and numerous day-trip options in surrounding towns. There are lots of relevant threads on this forum.
Naples is doable as a day trip from Rome, with fast train options. Much as I like Naples, however, the more interesting things to do in that part of Italy (IMHO) are outside Naples, and this region is best saved until you have the time for it.
In twelve days, some people will try and do the 'holy trinity' of Venice, Florence and Rome. Personally I think that option needs at least two weeks - Venice and the islands needs four full days.
If you're sticking to city centres, travel by train. There's no need for a car.
Fly 'open jaw', in to one city and out of another, to avoid wasting time back-tracking.