Attending mass in one of Rome's 900+ churches
Whether at St. Peter's Basilica or any of Rome's more than 900 churches, attending religious services is something most pilgrims and the faithful will want to do at least once—and frankly something every visitor to Rome should do, regardless of their faith (or lack thereof).
After all, in this most Catholic of countries, the foundation stone of the Christian church with 2,000 years of religious history and traditions dating back to St. Peter himself, attending Mass is just as important a cultural experience as it is a religious observance—especially a High Mass, in Latin with swirling incense and everything.
Maybe it's just my anthropology training, but I don't see how any visit to Italy is complete without going to a church service at least once. Catholic traditions have defined so much of Italian culture and history that it would be a sin not to sample this venerable aspect of Italian identity. (Yes, pun intended.)
You don't have to pray. You don't have to convert. You don't even have to understand what is going on. Just pick a pew and quietly observe. Stand, sit, and kneel whenever everyone else does (Catholic mass is a bit like slow-motion calesthenics). If you are lucky, there will be singing.
Just be respectful. This is not the time to wander around the church looking at Old Master altarpieces and snapping photographs.