Pu.t on your most comfortable pair of shoes and walk leisurely along the streets, piazzas and the river. Do not miss Ponte Vecchio. It is a joy to be alive and see it all.

You could walk around here forever. The atmosphere, the buildings, the statues, the sites, the history - amazing. You really feel like you are in a completely unique place in Florence - it isn't like anywhere else in the world. Love it.

I don't need to list the many museums and monuments one can visit and admire in the area, and despite the recent invasion of international brands (Zara, HM, etc) and the dominance of luxury brands (Gucci, Chanel, etc) there are still many small local designer shops, antique galleries and workshops (such as Martelli leather gloves via Por Santa Maria, Il Papiro paper goods via del Proconsolo, and Spezerie Palazzo Vecchio perfumes and toiletries just to quote a few) that make it worthwhile hunting for original gifts for loved ones or treats for oneself.
There are plenty of dining options too and the area is always pleasantly busy and very safe even late at night.

Just know that the shops in Centro Storico Florence are either very exclusive (Hermes, Chanel, Loro Piana, Gucci, etc.), or rather low end ("factory stores" with very poor quality workmanship). My experience is that the exact top-end merchandise can be found at Harrod's, Neiman Marcus, or Bergdorf Goodman and those stores put it on sale several times a year. Florence is NOT the place for bargains in shopping. BEWARE the knockoffs! It is a crime to export knockoffs from Italy. Fake designer handbags are sold on the streets by Nigerians. They scatter when the police arrive.
Enjoy the art, the culture, the food, the scenery. Shop elsewhere except for paper products like stationary.