When you start the dancing depends upon what type of reception you are having. If you are having a more formal affair with a sit-down dinner, you will want to wait until after dinner has concluded. Otherwise, start the dancing soon after guests have arrived and they have been greeted in the receiving line.
The first dance is traditionally between the bride and the groom. Song selection is often a large focus for the couple. Many couples pick a song with sentimental value; perhaps the one that was playing in the background on their first date. This dance can be anything the couple wants, however. Some couples opt for humorous choreographed routines, others just a traditional slow dance.
The second dance is usually reserved for the bride and her father, and eventually, the groom and his mother may join in. You don't have to have the parent dances directly following the first dance. At some point the wedding party should start dancing or the DJ can announce to the crowd so that guests know they may now join in the fun!