Because the city is compact and built on a medieval plan, streets are narrow and parking is scant. To see everything in Florence – or even a small fraction of everything – you'll need very good walking shoes.
Florence is widely acknowledged to be a veritable outdoor museum, and if you start from the old part of the city, you can walk almost anywhere and see beautiful monuments. Focus especially on the piazzas, which are often millenia old, and try to visit the following monuments especially.
Start in Piazza del Duomo, birthplace of Dante. Here is the Cathedral of Santa Maria del Fiore, also called the Duomo of Florence; this cathedral is one of the most awe-inspiring in Europe, with Brunelleschi's Dome, the famous Baptistery, and Giotto’s Bell Tower; you'll have to take pictures from the bottom looking up here for an incredible shot.
Crossing via Calzaiouli you arrive in Piazza del Signoria at the Fountain of Neptune, sculpted by Bartolomeo Ammannati to honor the Medicis (the face of Neptune is said to have been modeled on Cosimo), which is very much a symbol of Florence. Like many of the finest monuments in Florence, Neptune himself has been moved to a museum – the National Museum, in his case -- for preservation; the Neptune you'll see is a copy made in the 1800s.
Nearby is the most important Florentine monument, Michelangelo's David. Read the story behind it if you have a chance; this statue is the one that made Michelangelo's reputation and its making is as fascinating as the statue is magnificent. This monument is also a copy, with the original on display at the Academy Gallery. You'll also see the Ratto delle Sabine and Perseo by Cellini here.
From here, walk down the Via Vacchereccia, then the Via Santa Maria to the Ponte Vecchio, where you can enjoy this ancient bridge and all the shops that have been built up beside it right over the water.
This is a short walk; you'll find that nearly any walk from piazza to piazza will be rich in monuments. Bring extra memory for your camera.
There are several botanical gardens in Florence. The Giardino dell'Iris is a sight not to be missed if you are in Florence during May, when the irises, a symbol of the city, bloom. You'll find thousands of varieties of irises here, as well as a very fine view of the city and surrounding countryside.
Another wonderful garden is the Giardino dei Semplici, at the Via Micheli. This beautiful oasis is the third oldest in Europe, established in 1545 by Cosimo I de'Medici. Though it was originally designed to provide medicinal plants, it is today used for experimental gardening, and its central fountain, lovely walkways, and often ancient trees hold an air of antiquity you will rarely find even in Florence.
Entered through the Pitti Palace, the Boboli gardens provide a full day of walking beauty, from the amphitheater to the top of the hill that offers remarkable views of Florence. The gardens here will transport you to another world, one a long time ago.
Downstream of the Ponte Vecchio and also spanning the Arno, you'll find the Ponte Santa Trinita, named for a nearby church at via Tornabuoni. It's the oldest elliptic arch bridge in the world, and often said to be the most beautiful bridge in Florence. The Ponte Santa Trinita was destroyed by the Nazis, but the Florentines raised each stone from the bottom and reconstructed it in its original glory.
Other bridges of interest are the Ponte Alla Grazie, the Ponte alla Carraia, and the Ponte di San Niccolò. Strolling along the banks of the Arno river, and stopping over these ancient bridges, you’ll enjoy amazing panoramas of the Cradle of the Renaissance.