The climb to the top of Brunelleschi's dome is 463 steps, and generally takes 45 minutes to an hour in total (climbing up, viewing from the top, and climbing down). The steps were originally designed for construction workers, and there is no elevator.
Dress code: as mentioned earlier, this is still an active consecrated Cathedral and there is a strictly enforced dress code: chest and shoulders must be covered and pants or dresses must reach below the knees. Please follow the dress code, otherwise you may be denied access.
There's no view like it from any structure in the city. With no elevator, the 463-step climb gives you an up-close look at the Duomo ceiling and the surrounding marble, sandstone, and brick. The trek is worth its weight in gold. It's quite a climb to the top, but you must get the view if you can.
The Duomo rooftops offer a unique viewpoint and interact with the modern rising city. The terraces are accessible on foot, by climbing around 256 stairs, or with 2 elevators which take visitors up to the first-level walkways in a matter of seconds.
Admission to the cathedral is allowed only to people wearing clothing suitable for a place of worship. Visitors with bare legs and shoulders or wearing sandals, hats and sunglasses are not allowed inside the cathedral. Bulky backpacks and bags are not allowed.
If you're mostly interested in the views of the city, I'd recommend climbing Giotto's bell tower (get the BRUNELLESCHI OR GIOTTO PASS). You get amazing views of Florence AND the Duomo AND the Baptistry AND it's less crowded.
This is a bit of a smaller climb comparing to the Cupola one, only 414 steps and much wider. The bell tower has several levels so you can take a rest if you need to. The best photos of the Cupola and surrounds can be taken from here.
Does the dress code apply to the climb as well? As long as you are wearing no shorts and no sleevless tops, you will be ok. Same applies to all religious sites. I see long shorts (to the knee) in the Duomo in Florence and other churches all the time nowadays.
Thus site says there are 144 flights. That means 12.3 steps in an average flight, which seems realistic. Note that the flights narrow and shorten towards the top, due to the narrowing of the tower structure.
To enter the cathedral, visitors must be adequately dressed. All visitors to the cathedral are required to remove their sandals, sunglasses, and hats before entering. Avoid wearing low-cut tops, short skirts, and short shorts.
As you can see, it's pretty tight. Rick's description of the climb may be helpful: 463 steps up...the claustrophobic one way route takes you up narrow, steep staircases....