Hi Vince --- it's a GREAT idea!
My husband and I have spent 35 nights in Florence twice and a third time for 10 nights. We plan to do it again in January/February some time in the next few years, exactly to have fewer crowds than we experienced in September and October, and even in November. and December.
We've enjoyed all three neighborhoods we've stayed in: Sant'Ambrogio just west of Piazza Beccaria, Santo Spirito near the Basilica, and San Frediano just outside the city gates. I am especially fond of San Frediano.
We went on very few day trips, but you could go on lots by train or bus. My husband took both all day and half day Italian language classes at the Istituto Michelangelo and highly recommends them, so that may be something you'd like to do. The students are from all over the world and you can attend cultural activities, like an evening visit to the Accademia.
I took art history classes at the British Institute and loved them and still take some on zoom. Pricey. You could check out their free Wednesday lectures on zoom --- you can register here: https://www.britishinstitute.it/en/whats-on/wednesday-lectures. I'm not crazy about the lecturer this week, but in November are two lectures about the 1966 flood in Florence. You can also see their old lectures on youtube.
I feel like I don't need to give you a big long list of all the wonderful things to see and do in Florence --- you won't have any problem with making your own list. You can certainly make friends taking classes. Even I did, and I'm not very social and I speak no Italian. It's fun to find a coffee bar and a restaurant that you can return to again and again. There are several restaurants in Florence where you sit at the same table as other people and you can chat with them or at least not feel like you are totally alone.
The so-called "cold" in January and February (note where I live) will be 100 times better than how hot it is in the summer and fall --- my theory is always that you can add warm layers for cold but when it's boiling hot you can't take take them all off (and you'd still be hot, even if you did). The shorter days in winter don't bother me one bit at home and they are actually much nicer in Italy because the streets are all lit up and people are walking around enjoying themselves and dining late and shops are open. But I know that the darker days really bother some people.
What else can I tell you? The buses are both fun and useful, once you figure out their routes. The more you learn about Florence before you go, the more you'll find things you want to do, like on our latest trip I tried to find and photograph all the flood markers. So, yeah, your trip sounds like a great idea to me!