Florence in December

It's an incredible city. I didn't look up the December climate but I'd guess it's pretty chilly.

Definitely head to the Chianti Classico region and stay a night or two and check out vineyards. The towns I'd personally focus on there are Gaiole in Chianti and Greve. A little further south is Montalcino and Montepulciano. As a wine lover you should hit Montalcino for sure to get Brunello di Montalcino. It's one of the great wines of the world.

I'd contact Isodi if you want a small welcoming vineyard to tour and sample. One of my favorite vineyards in the region. The best bang for your buck is to do tasting at an enoteca which will allow you to sample a wide range of available wines from the region. The enoteca in Greve is world class.

I've been twice. Going in September for the third time and I can't wait to get back there. I'm hoping to spend extended time there one day. However long the Italian government will let me stay for a visit.

I’ve posted some of these recs in other threads on Florence....
—Nerbone: a lunch counter/stall inside the Mercato Centrale, in a far corner. Traditional Florentine tripe sandwiches, braised beef sandwiches dipped into the cooking juices, soups like ribollita, etc. Cheap, order at the counter, grab a nearby table, have lunch and people watch. The Mercato Centrale is a food market done only as the Italians can: everything is beautifully displayed, the stalls are gleaming and tiled.
—Il Guscio, in the Oltr’Arno/San Frediano neighborhoood across the river, via dell’Orto 19. Bistecca fiorentina....worth the price. You can even get it sliced over an arugula and parmesan salad, if you’re trying to balance out the pasta you’ve been eating.
—Cafe Rivoire, on the piazza della Signoria.....yes, it is overpriced because it’s on a piazza...but you can go inside and drink a quick espresso while standing at the malachite-green marble bar for a couple of euros, use the fancy & clean toilet, and people watch. Or, splash out like the ladies who lunch and sit at a table & watch the world go by.
—the Venchi store.....while Venchi are industrial and not handmade chocolates, they’re plenty good, and the store has a literal wall of flowing melted chocolate. You can smell it before you enter the store. Very tasty chocolate & coffe drinks, including a rum hot chocolate that would be lovely on a cold winter’s day.
—Santa Maria Novella Antica Farmacia; it’s a luxury toiletries brand, but the mother ship location near the convent of Santa Maria Novella is part museum, part store & very popular with the Japanese.

A general rule of thumb for Italy: if you can see a historic site from the cafe, keep walking, it’s likely a tourist trap. Yes, there are notable exceptions, but the local stuff usually requires walking away from the most touristed areas.