Florence has many other churches housing some of the finest art in the world: San Miniato al Monte, San Lorenzo, Santa Maria Novella, Santa Trinita, the Brancacci Chapel at Santa Maria della Carmine, Santa Croce, Santo Spirito, SS Annunziata, Ognissanti, and more.
Then there are the art galleries. The Uffizi and the Pitti Palace are two of the most famous picture galleries in the world. But the heart and soul of Florence are in the two superb collections of sculpture, the Bargello and the Museum of the Works of the Duomo. They are filled with the brilliant, revolutionary creations of Donatello, Verrochio, Desiderio da Settignano, Michelangelo, and so many other masterpieces that create a body of work unique in the world. And, of course, there is the Accademia, with Michelangelo's David, perhaps the most well-known work of art anywhere, plus the superb, unfinished prisoners and slaves Michelangelo worked on for the tomb of Pope Julius II.
In all, Florence has something over 80 museums. Among those at the top of most lists are the City hall, the Palazzo della Signoria (aka Palazzo Vecchio), a wonderful building with magnificent rooms and some great art; the Archeological Museum, the Museum of the History of Science, the Palazzo Davanzatti, the Stibbert Museum, St. Marks, the Medici Chapels, the Museum of the Works of Santa Croce, the Museum of the Cloister of Santa Maria Novella, the Zoological Museum ("La Specola"), the Bardini, and the Museo Horne. There is also a wonderful collection of works by the modern sculptor, Marino Marini, in a museum named after him. If you are interested in photography, you should not miss the superb collection of works by the early photographers, the Alinari brothers. The magnificent Strozzi Palace is the site of many special exhibits.
To get a great overview of the city, you have plenty of choices: climb the dome of the Cathedral or Giotto's Bell Tower which is much easier or head for Piazzale Michelangelo a large parking lot on the hillside just south of the center of town, or climb a bit further to the church of San Miniato al Monte, a sublime 11th century masterpiece, with superb Renaissance sculptures. At vespers, the monks add to the beauty with chants.
There are also a few places to buy things, from the high-end jewelry stores lining the Ponte Vecchio to some of the most famous shops in the world; Gucci, Pucci, Ferragamo, Valentino, Prada, Armani, Ermenegildo Zegna, Buccellati, Frette, as well as many more wonderful shops that aren't yet world famous. It is increasingly difficult to find bargains, but keen-eyed shoppers can still find good deals on smaller, side streets running off of those above and elsewhere in the center of town.
Great places to walk include along the Arno and across any of its bridges; through narrow, medieval back streets in the Santa Croce area; and in the Oltr'Arno - on the south side of the river, in many ways like Rome's Trastevere or Paris's Left Bank - but far, far smaller. There are also superb shopping streets, such as the Via Tornabuoni, the Via del Parione, and the Via Maggio.