As far as I can tell, entry to the Duomo is not included in the Firenze Card.

You can't book tickets for the Duomo; it's free. Queues are very long.
You can buy timed tickets to climb the dome; I don't think this is included in the Firenze card.

Actually, I think the climb to the Cupola is included, but the reservation process is somewhat convoluted. From the Firenze Card website:

"In order to make the Cupola reservation it is necessary to collect at first the Firenzecard; go to the Grande museo del Duomo ticket office (piazza S.Giovanni,7); collect the Grande museo del Duomo combined ticket and, at the same ticket office, make the Cupola reservation at the dedicated screen. Please note that, if you have not activated your card yet, the collect of this ticket entails the activation of your Firenzecard and that you have 72 hours from this moment to visit the museums."

I don't think it's worth waiting in a long queue to see the inside of the Duomo. There's not much particularly interesting about the architecture, apart from the dome. There's very little art inside; most of it's now in the Museum of the Duomo, which is well worth a visit. The Baptistery is also very beautiful inside. Another very beautiful church, inside and outside, is Santa Maria Novella, near the train station.