I am in the North and its already getting cold, by January you might find driving treacherous or even impossible in the most interesting parts.

I would suggest something like the following, driving South to North:

1. Fly into Bari or Catania. I've found Sicily is every Italian's second favourite destination and Puglia is every Italian's second favourite cuisine (after wherever they are from, of course).
2. Naples and the Amalfi Coast. Naples is the most quintessentially Italian city but i wouldn't recommend driving into the centre unless you take the full insurance package on the car.
3. Rome. There's nowhere on earth that comes remotely close to the history and culture there. You could spend the whole week here and not see everything.
4. From here, and weather depending, either go over the Gran Sasso Park and take the long way to Bologna, or go straight up, stopping in Siena or Florence on the way.
5. If cars are your thing, spend a day touring the Ferrari, Lamborghini, Maserati, Ducati, or Pagani factories, all of which are within a few miles of Bologna.
5. Fly out of Bologna.

That should be plenty for a week's driving, I wouldn't plan for more than 2 or 3 places in that time. I also wouldn't bother with Milan or Turin in winter, and Venice is pretty boring.