Getting to know Wine & Distilled drinks in Florence

Two places that come to mind, both within a half block of the Central Market and within a block of each other, are Casa del Vino and Zanobini. Casa del Vino would be best visited for tastings after their busy lunch period. Zanobini is purely a wine shop, and they close for a few hours in the afternoon. Both have onwers very knowledgeable about Tuscan wines. Don't specifically recall whether they have local spirits available for tasting, but I suspect at least one of them does.

The Golden View bar, named for its great view of the Ponte Vecchio, now has a tasting menu. They are already well known for offering a pretty ample buffet of appetizers for aperitif (cocktail hours) along with a wonderful view. We found they were very friendly there as well.

http://www.goldenviewopenbar.com/en/wine-tasting/

Cantinetta dei Verrazzano (via dei Tavolini) is a good place to eat fresh appetizers and taste wines from the Verrazzano grounds. Not a bad place to have a simple lunch or dinner but go early as sitting places are limited in number.

Enoteca Alessi (via delle Oche) has a tasting space as well and has the largest wine choices in town.

There aren't really local distilled drinks. Occasionally grappa may be made from Chianti but it is an imported custom. The traditional way to end a dinner in Tuscany is over sweet wines - Vinsanto or the rare and very expensive Aleatico. Alessi may sell spirits from all over the world but there is almost no local production.

However you do it, try a Brunello di Montalcino.