There's two main ways to see the Holy Father: the Wednesday audience and the Sunday Angelus/Regina Caeli.
The Wednesday Audience: Nearly every Wednesday, the Holy Father has a public address in St. Peter's Square or in the Paul VI Audience Hall, if the weather is bad. The address starts about 10AM and lasts about an hour. There will be some short reflection by the Holy Father in Italian, but there will be synopses given in English as well as a host of other languages. At the end of the audience, the Holy Father also gives his blessing; it's here that you can bring all your religious objects to be blessed by the Holy Father. At some point, either before or after, the Pope will make a trip around the Square in the Popemobile. If you're on an edge, or more preferably a corner, you'll get a good shot of the Holy Father. Know that you'll likely spend the better part of your morning here as it's good to arrive at the Square around 7 or 7:30AM. Also, you'll need tickets to get in; the good news is that they're free. Just email the US Bishops Visitors' Office in Rome at: visitorsoffice@pnac.org. If you're coming as newly-weds, you can get special seats. Just be sure to ask the Sisters who run the Visitors' Office for details.
The Sunday Angelus/Regina Caeli: Nearly every Sunday, the Holy Father gives an address at about 12noon from the window of the Apostolic Palace. Just gather in St. Peter's Square about 11:30 or 11:45AM to get past the security; there are no tickets…just show up! The address only lasts about 10 minutes during which the Holy Father gives a short reflection on the Sunday Gospel. Depending on events in the world, he may also make some short corresponding comment. After this, he will pray the Angelus or, if it's during the Easter Season, the Regina Caeli. Either prayer is recited in Latin. The Pope will likely conclude by saying, "Buona domenica" and "Buon pranzo", both of which are typical Italian greetings. The first means "Happy Sunday"; beautifully, it's the only day of the week which has a special greeting attached to it. The second means "Happy Lunch" or "have a good lunch"; it's usually a friendly greeting shared with friends and/or with complete strangers. Either way, it's rather endearing.