Second-hand and rummage aren’t quite as big in Italy as in many countries but there are options. The biggest shops are out of the center. Churches also often have temporary rummage markets around holidays.
Second-hand shops
Clothing and furniture
Di Tutto di Più — Via dei Vanni, 56/a
Mercatopoli Legnaia — Via Frà Diamante, 12/a
La Mecca dell’Usato — Via Poggio Bracciolini, n. 8/14
Nei Panni degli Altri — Via dei Macci 11
Just clothing
Charity shops for Progetto Arcobaleno — Via Leone 28r and Via Palazzuolo 9/r
Occasional Markets
Le Cascine — every Tuesday and Sunday, an enormous market with everything in the world, new or used
Only Usato — once monthly, currently at Il Conventino
Online options
Te lo regalo... — Free things if you pick up
Vintage

Where to donate used goods
The municipality of Florence has volunteer networks that take donations. District 1.
Donate to Caritas in yellow bins
H&M clothes recycling program — drop off at store, including damaged clothing
Charity shops for Progetto Arcobaleno — Via Leone 28r and Via Palazzuolo 9/r