We were not so dumb as to go right from the Duomo climb to the campanile! We took a break and went to the Opera Duomo Museum, which is behind the Cathedral of Santa Maria del Fiore (the Duomo). This was new to me, and was fabulous. Ghiberti's original baptistry doors are displayed here, as is one of Michelangelo's last pietas and many of the panels that adorn the exterior of the cathedral.
This was a great museum. Don't skip it if you go to Florence - it's part of the 15 euro ticket you can buy that gives you access to the duomo and the campanile.