Take the train...seriously.
For your Florence itinerary--I did Florence in one busy day in December and we fit in: Santa Maria Novella, Duomo & Bapistry, lunch, Ponte Vecchio, Uffizi, Accademia (for David), and dinner. We tried to fit in San Marco but it closes early. Things in Florence do tend to run on a weird schedule, so I'd recommend verifying opening/closing times if you can (and even then they don't always stick to them!)
If you want to see the Uffizi, reserve. I liked Santa Croce (I saw it on a return visit to Florence) but it's pretty heavily damaged from floods. On the other hand, that's where Galileo, Michaelangelo, Machiavelli and others are buried. Bargello has sculptures while San Marco has frescos.
Must-see depends on whether you are more interested in art, history, churches, or just wandering around and seeing the city.