· St. Peter (San Pietro): The one everyone knows because it's the focal point of Vatican City. Built over the site of St. Peter's tomb, the basilica is the main spot for large Papal masses and tourists can pay to climb the winding stairs of the dome to get a view over Rome from the tippy top cupola.
· St. Mary Major (Santa Maria Maggiore): Completely dedicated as it is to the Virgin Mary, you'll find a ton of feminine imagery throughout the artwork and stained glass. Only important holidays mean the use of the entire church; normal daily masses happen in the side chapels, for a more personal experience.
· Archbasilica of St. John Lateran (Arcibasilica San Giovanni in Laterano): If you only visit one of these four, it should probably be this one because...deep breath...it's the Pope's true seat. That's why it gets the fancy title of "Archbasilica." St. John Lateran is historically the home church of the Pope, not St. Peter's as most would believe.
· St. Paul's Outside the Wall (San Paolo fuori le Mura): Just as St. Peter's was built over the burial spot of St. Peter, so St. Paul's is over the burial location of St. Paul. This also explains why it takes a little longer to reach; it's on the road to Ostia, accessible by subway line, but not in the center of Rome. If we had to pick a favorite basilica, it'd be this one for the main reason that the monks here still produce their own booze, all for sale in the gift shop. Try the eucalyptus liquor!
Once you've visited all four (supposedly with reverence and humility), have fun with your shiny new indulgence. You don't get anything physical to show for it, save for the promise from the Catholic Church that one of your sins will be swept under the rug and forgotten.