I have been on top of the Dome a few times as well as Giotto's bell tower next door, the roof of the Palazzo Vecchio and once the Leaning Tower of Pizza, all a long, long time ago, happy, happy days!
The Dome is a true self supporting structure, unlike the Dome of St Pauls in London, a marvel of engineering and no lives lost in its construction.
To get to the top you have to climb winding steps inbetween the inner and outer walls of the dome, near the top they get very, very steep and lean a lot sideways, would love to climb them again but alas I think it would be a bit beyond me.
I can recommend the book "Brunelleschi's Dome" about its building and the man who designed it, it's based on the shape of an egg shell!
P.S. When I climbed The Leaning Tower in the sixties it was all open including each of the platforms as you went up, no guard rails at all, all completely open, nowadays Health and Safety would throw a fit seeing that!