As far as leather goes it depends on what quality you are after.
Many on here think the market stalls are a good place to shop, but there are many different qualities and definitions of leather, and I find lots of ths stalls not to my taste for a special purchase.
People do say the leather school is a good place - I have no experience of that.
My husband bought me a really lovely bag here:
https://www.thebridgefirenze.com/
via Vacchereccia, just off the bottom of the Piazza della Signoria
Despite the name, they are Italian handmade bags, not cheap, but not Gucci type prices.
Tjere is also a really good gelateria and couple of doors down.