Genoa area early January

callico,

Many of the tourist destinations in that area will probably be "quiet" in January. Some may be closed or operating on reduced hours.

At that time of year, I wouldn't spend six days in Genova, as your time would be better used elsewhere. You might consider travelling by train to Florence (about 3.5 hours with one change) and spend the bulk of your time there, using it as a "home base" (Florence is the capital of Tuscany). From there you can easily take trips to Siena (use Bus from Florence), Lucca (train or bus), Pisa (train) or other locations. You could also explore some of the historic sights in Florence such as the Uffizi or Accademia.

Check your local Library to see if you can find a copy of the RS Italy guidebook, as there's an enormous number of suggestions in the books on sightseeing, getting around Italy, hotels and restaurants.

We enjoyed Torino in winter twice. Many great museums and generally very good weather at least when we were there. Makes a good base for day trips into the Piemonte wine country to towns such as Alba and Cuneo, and Milan is only an hour away.

Another "shout out" for Torino. With its covered arcades and cozy cafés, the city is built for the winter, and deserves 4 nights (this allows for a day trip somewhere). You can then spend 2 nights in Genova to explore (I challenge the fact that the city is "expensive", by the way - especially in January: did you actually check hotel prices?).

I don't recommend it over Florence or Torino, but Bologna is another city with an historic district full of arcades, making it comparatively pleasant on days with precipitation.

Yes in January you want to base in a city and then depending on the weather do day trips to more rural areas. Genoa itself could warrant a couple days, but most of the places you would day trip to from Genoa as a base are coastal areas that will be largely 'closed' up and definitely not fun if you get bad weather. Florence is a good suggestion as there are tons of things to do inside (churches, museums) if bad weather plus it's a great base for many other towns (Siena, Lucca, Pisa, smaller hill towns) if you get some good weather. Same would be true for Bologna (day trips from there include Modena, Ferrera, Parma, Ravenna). In fact Bologna and Florence can be visited as a day trip from each other. I think Torino is a less good suggestion because, like Genoa, there isn't quite as much to keep you busy for 6 days and the day trip options would be more weather dependent. But it's a nice city for a couple days.

I could spend a week in Torino and never get more than 10 miles away there is so much there!

Take a train to Florence upon arrival and use Florence as your base for Tuscany as suggested above.
Return to Genoa a day or two before your ship departs. One day in Genoa is more than sufficient to see that little there is to see of a not so pretty industrial port city.
Most things to see are in or near the ancient port, namely:
La Lanterna (3rd oldest still standing lighthouse in the world)
Carruggi (alleys) of the historical center
Galata Maritime Museum
and, if you are interested:
Aquarium (largest in Europe)
Biosphere

Genoa is Christopher Columbus’ birthplace. However if you plan to visit his birth house I want to warn you that house is a copy of the original. The original house was destroyed by the French in the 17th century together with many other parts of the city.