The historical center, Centro Storico, is a great place to start this portion of your Europe tour!
The Piazza de la Signoria, nearby Palazzo Vecchio, Ponte Vecchio (the amazing historic bridge over the river Arno), Cathedral Square (Piazza del Duomo) and Boboli Gardens are the (predictably?) unmissable Florence attractions. Who can argue?
In fact, in this most pleasant of cities for pedestrians, all these Florence sightseeing highlights are within walking distance from each other and can be seen in a single day for those short of time. However, a week minimum might be a better idea for lovers of art, which will permit to take in the rich historic atmosphere of the city at their leisure.
One possible idea is to actually stay at the historic centre, then have short walks to the different attractions between the extremely tasty meals you’re sure to find in this extraordinary city.
A moderately-demanding walk is to go round the river Arno in a circle, i.e. crossing at the furthest bridges (that you wish to choose) apart. This will permit an excellent appreciation of the city’s architecture and sheer atmosphere, you’ll see some locals and escape the tourist hordes! This walk can take 2-5 hours, depending on how many detours you choose to make! Also, try visiting the hilly end of the city, you’ll get fantastic views and a great workout!