Audiences with the Pope are held just about every Wednesday morning. As you can immagine, tickets need to be reserved well ahead of time. There are a couple of easy ways of accomplishing this.
Arranging a Papal Audience by Telephone
You can arrange an audience with the Pope through the Prefecture of the Pontifical House of the Vatican City:
Arranging a Papal Audience via a Web Form
Santa Susanna, home of the American Catholic Church in Rome, offers a means of aquiring tickets to Papal Masses and Papal Audiences
by filling out a web form well in advance of your trip to Rome. They will attempt to make your tickets available for pick up at the church the evening before the event.
Although rarely available, you may be able to reserve a Papal Blessing
Vatican Dress Code
Be aware that Vatican dress code requires no shorts or tank tops. Women's shoulders must be covered.