With some planning, you can see Florence's most famous sights in a couple of days. Start off at the city's most awe-inspiring architectural wonder, the Duomo, climbing to the top of the dome if you have the stamina (and are not claustrophobic: it gets a little tight going up and coming back down). On the same piazza, check out Ghiberti's bronze doors at the Battistero. (They're actually high-quality copies; the Museo dell'Opera del Duomo has the originals). Set aside the afternoon for the Galleria degli Uffizi, making sure to reserve tickets in advance.
On Day 2, visit Michelangelo's David in the Galleria dell'Accademia —reserve tickets here, too. Linger in Piazza della Signoria, Florence's central square, where a copy of David stands in the spot the original occupied for centuries, then head east a couple of blocks to Santa Croce, the city's most artistically rich church. Double back and walk across Florence's landmark bridge, the Ponte Vecchio.
Do all that, and you'll have seen some great art, but you've just scratched the surface. If you have more time, put the Bargello, the Museo di San Marco, and the Cappelle Medicee at the top of your list. When you're ready for an art break, stroll through the Boboli Gardens or explore Florence's lively shopping scene, from the food stalls of the Mercato Centrale to the chic boutiques of the Via Tornabuoni.