The Duomo is worth going inside, but I wouldn't spend too long. Most of the intresting stuff has been removed and placed in the Cathedral Works Museum (definately worth seeing), which is across the street from the back of the Duomo.

As for climbing to the top of the Dome, if you want to do it, I'd suggest climbing the Campanile instead if it has a shorter line. The view is about the same.

The Baptistry in front of the Duomo has a beautiful ceiling.

Santa Croce is interesting. A lot of tombs in there - its kind of a Who's Who of Italy in a sense.

As for the Uffizi Gallery and The Galleria Accadmia (with the David and the Slaves), there both incredible. There a phone number you can call for reservations, which I would highly recommend. I can try to dig it up for you if you'd like.

There's a science museum right behind the Uffizi that's a little off the beaten path. Lots of interesting stuff on Galileo and astronomy. It also has a very cool collection of old maps and globes.