Your correspondent climbed both the Campanile, the bell tower, and the Duomo itself on successive days and happily felt none the worse for wear. She enjoyed the views and the help with orientation that only a bird's eye view can supply.
The first great church in the near distance is Basilica di Santa Croce, where Galileo, Dante, Machiavelli and Michelangelo and many other historical figures rest or, for some, are at least forgiven and remembered, with monumental sculpture.
The second monumental building is City Hall. Between it and the River Arno is the Uffizi Gallery.
In the distance is another large church, San Miniato al Monte.