To join a family you have to be invited by one of their top3 (Don, Sottocapo or Consiglieri). You can contact them ingame or via IRC, families use to have a recruitment channel. Check the family page to know it top3 and maybe its public and recruitment channel.

Recruitment Centre

This is a very good place to start to look for a family!
On Recruitment Centre you can see the 20 families with the most available space in the game and you can see how many free space each of these families has.
Also, here you can see the Don and the Sottocapo of each family.