In the old town center: 
Vecchio Bridge
Santa Maria del Fiore , also known as the Duomo di Firenze is the city's beautiful cathedral, the symbol of the city. Brunelleschi's huge dome was an engineering feat of the rennaissance. A statue of Brunelleschi is sited in the piazza, with his figure looking upwards towards his dome. It is possible to climb the Dome (entrance on the side of the church), which has 464 steps. ?8 entrance fee (June 2010), and usually has a long lineup.
Giotto's Tower- adjacent to the Duomo, you can climb the tower for a magnificent 360-degree view of the Duomo, Florence, and the surrounding area.?6 entrance fee, and requires some tenacity to climb 414 steps.
Baptistery famous for bronze doors by Andrea Pisano (14th century) and Lorenzo Ghiberti (15th century) and a beautiful interior the vault of which is decorated with 13th century mosaics (the only medieval set of mosaics in the city.
Palazzo Vecchio - old city palace/city hall, adorned with fine art. The replica of Michelangelo's "David" is placed outside the main door in the original location of the statue, which is a symbol of the Comune of Florence. The site displays an important collection of Renaissance sculptures and paintings, including the Putto, by Verrochio, and the series of murals by Giorgio Vasari at the Salone dei Cinquecento (Hall of the Five Houndreds) - the hall which used to display the now lost Renaissance masterpiece, that is, the so-called Battaglia di Anghiari, by Leonardo da Vinci.
 
Galileo's Tomb, Santa Croce
Ponte Vecchio the oldest and most famous bridge over the Arno; the only Florentine bridge to survive WW2. The Ponte Vecchio (literally "old bridge") is lined with shops, traditionally mostly jewellers since the days of the Medici. Vasari's elevated walkway crosses the Arno over the Ponte Vecchio, connecting the Uffizi to the old Medici palace.
Santa Croce church contains the monumental tombs of Galileo, Michelangelo, Machiavelli, Dante, and many other notables in addition to artistic decorations. There is also great artwork in the church. And when you're done seeing that, a separate charge will gain you admission to the Museo dell'Opera di Santa Croce, where you can see a flood-damaged but still beautiful Crucifix by Cimabue (Giotto's teacher), which has become both the symbol of the flooding of Firenze in 1966 and of its recovery from that disaster. The Pazzi Chapel, a perfectly symmetrical example of sublime neo-Classic Renaissance architecture is also worth visiting.
Santa Maria Novella, near the train station, is a beautiful church and contains great artwork, including a recently restored Trinity by Masaccio. Also, the Chiostro Verde, to your left when facing the front entrance of the church, contains frescoes by Paolo Uccello which are quite unusual in style and well worth seeing, if the separate entrance is open. Off of the church's cloister is the wonderful Spanish Chapel which is covered in early Renaissance frescoes.