January has snow issues for driving, but it would be a truly special experience to see Venice in snow, which IIRC doesn't happen every year.

Many car rental companies offer cheaper weekly rates so you get something like 7 days for the price of 5. It would be good to coordinate with this and only do the car hire for seven days, which means staying in a city at the start or end of your trip.

Venice is a prime candidate as you don't want a car there, but you probably don't want a car either in major cities such as Rome, Florence,.....My advice would be to fly to Venice, spend a few days, then pick up the rental car and focus on somewhat smaller cities but avoid the mountains. Bologna is an amazing historic site while Modena has amazing food. Verona is also very worthwhile, although you could also focus on smaller towns and country hotels such as the Italian members of R&C. Alternatively, from Venice you could go south along the coast and see the mosaics plus San Remo--I'm not sauté this is worth more than a day or two, plus in low season you don't need to escape from the crowds--or northeast into prosciutto country (where you'll also find modest restaurants specializing in steak on a wood grill and local wine which might be Cabernet franc). Be careful about taking a rental car over the border, for example into Croatia, as your insurance is unlikely to work here, plus you would need an international driving liscence.

Alternatively you could focus on Tuscany, visiting a subset of Florence, Pisa, Lucca, Sienna, San G, etc.

Yet another strategy is to go south but avoid Sicily, so that you would fly into and out of Naples (it could lower rental car prices if you return it to the same location) and do the Amalfi Coast plus Naples (and the volcano) in very low season, including perhaps Capri or Ischia, but check what's open at this time.

IMO even central Naples is a bit scary and I'm not sure I would want to attempt Sicily without knowing what I'm doing......

Also remember that Liguria can have floods and landslides in winter too, train in Italy between the main sites is always a good option

Another vote for Sicily here and flying in and out of Catania. More TP for CE too. Amazing fish/food market. Head south to Siracusa/Ortigia for more great food at Fratelli Burgio.

Eating a cannoli right now and pining to go back to Sicily!

....and remember that in Italy, one is supposed to drive on the right side of the road. If you've never driven in continental Europe or the USA, please don't attempt too much too quickly as many of one's instincts become counterproductive. It's even harder when you're trying to navigate in a foreign language unless you're accustomed to the standardized European road signs and the way of following a route by going to the next city in the sequence along the way (versus following a road number or signs toward the really big city which is your destination).

I did exactly this last year and it was amazing.

LCY - FLR definitely easiest and gets you into the thick of the Tuscany action quickly without messing about in Pisa as you would if flying from LHR or LGW.

Our route was something like this:
drive down to Siena, stay in an agriturismo near the city for 2 nights.
drive down to Montepulciano area, 2 nights in another agritusmo round there
drive up to Greve in Chianti area, stay on a vineyard for 1-2 nights
drive into central Florence, return the car and walk to hotel. Stay 1-2 nights. Florence is wonderful but felt like a relative dump compared to the other amazing places mentioned above.

Thoroughly recommend the trip, very easy and amazing scenery. And the food and wine...

If you can drive in London, Sicily will hold no terrors for you.