Italy in January

Just remember the days will be short, potential damp and cool.

Not January, but February. Cinque Terre is really not a winter destination. Many places closed. The trails may be closed too due to weather.

Not much will be open in Cinque Terra in January, and the weather can be cold and wet. Don't think you'll find much to do.
I'd suggest Rome, Tuscany, Florence, and Venice. However, if you travel this route, fly into Venice and out of Rome.
What are your interests?

The Christmas season ends January 6th so depending on when you come you might see some holiday decorations, activities etc. It is usually cold and often rainy in January and February. As mentioned, it is not the time to visit Cinque Terre since a lot of things will be closed and some paths won't be accessible. Otherwise, just look and see what you would like to visit in Rome and Tuscany. Enjoy your first trip.

It's true as others have mentioned that the weather is unpredictable but we spent a week one January in La Spezia and day tripped to Cinque Terre - it was a splendid sunny day and not busy as soon as we got outside the main village centre. I suggest you keep it on your list (northern Italy in the depths of winter can be drier than later in spring) with a back-up option if the weather is bad - Lucca, or Pisa for example. You can check which trails are open a few days before you go, but several were when we visited in January - Cinque Terre is popular with Italians at the weekend even in winter because it's warmer than the rest of northern Italy.
This was our day January 2020: https://slowtravelitalyspain.blogspot.com/2020/02/la-spezia-in-january-manarola-to.html
The whole week the weather was decent - that's why many Italians escape from Milan to Liguria after Christmas, places like Lerici are very busy.
January (make it after 6th January so schools have gone back and there will be nobody around) should be very nice in Rome - perhaps the only 'quiet time' of the year left. You should also be ok in Tuscany if you keep to the larger cities like Florence and Siena, although Florence is colder than Rome in winter. Instead a friend told me that in smaller towns and villages in Tuscany which attract tourists the rest of the year many restaurants and bars are closed.
Venice is 'unmissable' but it could be very cold and damp.
Wherever you go, north or south, make sure your accommodation has got 'proper' heating, not just aircon.
I used to be sceptical about travelling in Januray but spent a month in Spain this year from 10th January and it was fantastic.
Also since everything will be quiet you could line up some suitable accommodation and then decide according to the weather forecast.

Rome can be delightful in January, as the city seems to relax after the January 6 end of the Christmas season, and although the days are short, some of the brightest blue skies I have ever seen there were in January. It's a great time to stroll around the city without encountering huge crowds.

I just HATE how early it gets dark at that time!

I agree with folks saying you need some backups. I did CT in the end of October I think? And it was rainy and not as picturesque as it would be if it were sunny. It wasn't overly enjoyable.

I've been to Rome in January a few times (great sales after the 5th) and weather has almost always been great....walking around in shirts in mid-day. Once we had snow for about a minute but that cleared quickly. I thought it was a great time to be in that city. Venice in January, windy and rainy when I've been there. Tuscan hill towns were cool...but nice weather.

IMHO January is a great time to visit Italy as there are not crowds around, but remember we are in winter. Your activities should be city centered, and you won't have big problems with short daylight hours or bad weather. You can sometimes find very clear mornings with good weather, but it won't be summer and you must have a backup plan if there is rain or weather is cold.