In Italy, having breakfast means eating sweet food and drinking cappuccino (by the way, breakfast is the only time during the day when you’re supposed to have a cappuccino). Deal with it. Once you’re eating sweet food, nothing is better than sweet pastry and, once you’re eating sweet pastry nothing (in Florence) is better than Giorgio. Giorgio, making Florentines’ life sweeter since 1972, is an institution in its field.
They produce and serve high-quality pastry all the (working) time, but the best time to go there is in the morning, especially between 07:30 and 09:30, when local workers, students and employees are having breakfast. I still remember spending so many mornings there back during high-school, having breakfast with some class-mates while skipping school. I still remember the fear of running into a professor (since Giorgio was very close to the school), but we couldn’t help it, we had to have our perfect breakfast there despite the high chance of meeting a teacher (luckily, it never happened).
They also serve a very high-quality aperitivo for a reasonable price (12 €), but it only lasts until 20:30.
They also serve an extremely good aperitivo starting from 18:30.
Last thing: Giorgio is not in the centre (not even within walking distance), it is situated in a nice, tourist-free neighborhood and you can easily get to by bus (number 6, from the main train station).