Martha,
I don't recall whether tickets are required for the Duomo, however they are required to climb the Tower which is next to the Duomo (lots of steps and circular staircase, but a fanastic view!). There's also a Baptistery around the corner and downhill (somewhat in the "basement" of the Duomo).
One reminder though - they're strict about the dress code in the Duomo so no bare shoulders, etc. or you'll end up wearing the dreaded "blue paper tablecloth". The Duomo is magnificent inside and well worth seeing.
"Il Campo" is definitely a worthwhile stop! There are lots of interesting small Cafes and Gelato shops in the surrounding side streets. If your timing is right, you may be treated to one of the local celebrations in Il Campo, with the participants dressed in the clothing of their contrade (I witnessed a "spettacolo" on my last visit with a huge awards ceremony for the best tamburino). The Church of San Domenico is also very impressive (as I recall, a bit more restrictive though so photos were not allowed inside).
Check the Italy Guidebook for restaurant listings, as there are some good restaurants there.
Happy travels!