Which museums should I book for?

Definitely book for the Uffizi. The Uffizi Gallery always has longest lines especially in the high season. In the dead of winter you'll find you probably won't need to worry about a booking but even in October it is a very good idea to reserve tickets ahead to save time, especially as you don't have much time in general.

Michelangelo's statue of David can have very long lines in too, so if he is on your list, avoid disappointment and book tickets for him too or try to visit late in the afternoon, when the lines begin to shorten significantly (the museum is open until 6:45pm).

The other sight you need to book for is to see the fresco cycle of Masaccio in the Brancacci chapel (Santa Maria del Carmine). This frescoes are protected by only allowing a small number of people in the room at once, and for only a maximum of 15 minutes at a time. It's not a busy crowded sight, but bookings are strongly recommended for entry.

Other museums such as the Bargello and the Museum of the Duomo have beautiful works by Michelangelo and Donatello but mysteriously don't have the volume of visitors that the other museums have, so there's no need to buy a ticket in advance for these. Just watch out for the alternating opening times on Sundays and Mondays at the Bargello!