Visiting the Vatican
The Vatican, both a tiny independent state within the city of Rome and the head of the Roman Catholic Church, has lots of interesting stuff for religious and non-religious alike.
The Catholic Traveler’s Guide to Rome - one of the best references I've found on the internet
...also see the Wikinapoli Kids in Rome page for tips on how to make the Vatican interesting for kids
The Vatican Museums and the Sistine Chapel
The Vatican Museums are a series of galleries that showcase some of the best paintings and sculptures in Italy, plus Michelangelo's famous ceiling and Last Judgment frescoes in the Sistine Chapel. In the summer, there are often long lines for the Vatican Museum. To avoid the lines, either get there early, book online (€4 booking fee for each group of tickets), or book a tour group (since groups skip the line). Open M-Sat 9-6, last ticket sold at 4. Open and free (and thus crowded) on last Sun of the month, 9-2. See the main Rome page for recommended tour companies.
Consider going at night (usually offered late April - late October on Friday nights), RomeWise calls this "one of the best-kept secrets in Rome" since it's much less crowded.
Various other "special" tickets to the museums are available on the ticket website, like a "Prime Experience" with a tour 1 hour before opening followed by an American Breakfast (this Rick Steves forum discussion notes that this allows you to see the Sistine Chapel before the crowds descend on it). "Hidden Vatican Museums" mostly makes sense for groups (since there's a €300-500 fee to "open" a space, plus additional fees).
Other ways of beating the crowds are to go on Wednesday morning (since many people are at the Pope's General Audience), go mid-afternoon (when many tour groups are at lunch), or go in the evening before closing (although the guards will shoo you out 30 minutes before the closing time).
St. Peter's Basilica
St. Peter's is both impressive and beautiful; Michelangelo's design and dome are outfitted with a panoply of religious art, including Michelangelo's own Pietà. Open daily 7-7, closes at 6 Oct-March. Note that the dress code is strictly enforced: no bare shoulders or shorts are allowed (yes, they will stop you). If you are up to it, don't miss the climb to the top of the cupola (the top of the dome), which gives you a great view of Rome and a different perspective on Michelangelo's impressive dome--from the inside (€6 ticket, 8 to 5:45 or 4:45 from Oct-March). More info at the Saintpetersbasilica.org Travel Information page.
Since 2017, there have been very long security lines during peak tourist season; AnAmericanInRome gives a good summary of how to skip these lines (apparently, the old trick of going directly from the Sistine Chapel to the Basilica doesn't work anymore unless you're with a tour group). The Catholic Traveler recommends just getting there super early (~6:45 AM just before they open), because then you can not only avoid a long line, but also experience the church without crowds.
Seeing the Pope
Wednesday General Audience
Held on Wednesdays at 10:30 AM (sometimes 10 AM in summer) in St. Peter's Square
Usually consists of prayers and a short homily (in Italian and Latin)
PapalAudience.org isn't an official site but has an easier-to-read summary schedule
Anyone can pass by and watch, but the best seats/views are with a ticket
reserve tickets well in advance!
Vatican prefecture website is the official source
Sometimes last-minute tickets are available from the Swiss Guards the day before between 3 and 6 PM (7 PM in summer) at the Bronze Door Entrance (just after the security line)
Americans will likely pick up their tickets at the Pontifical North American College (near the Trevi Fountain)
Sunday Angelus Blessing
Held on Sundays at noon in St. Peter's Square
Shorter blessing (in Italian and Latin)
Can also be cancelled for summer vacation/travel
No tickets given/required
The pope also presides over major services in St. Peter's (or the square outside). For example, there are papal masses on Easter Sunday and Christmas Vigil (midnight mass) each year. All of these require tickets; see the ticket information above for details. (Reports are that the Christmas mass is lots of fun—bring a blanket and some wine to make the wait outside more bearable!) See the Pope's Schedule for a full list.
Other Religious Events and Masses
St. Peter's has a number of Masses (not led by the pope) offered pretty much every day. The official schedule is here (in Italian, "festivi" means Sundays and holidays, "feriale" is other days), although St Patrick's or The Catholic Traveler's translations are easier to follow. Some of the services are (partially) in Latin, but mostly they are in Italian; see the Catholic Mass in Italy page for tips on how to follow along.
A useful resource for religious tourists is St Patricks Catholic American Parish (formerly at Santa Susanna), the home of the American Catholic Church in Rome. They offer English masses and help English-speaking pilgrims (especially Americans) find their way. The priests lead occasional religious walks for English-speaking Catholics in Rome (you can also find videos of former pastor Fr. Greg showing Roman sites from a religious perspective). Their website also lists convents that offer cheap rooms for pilgrims
Other Special Tours
The Vatican offers a few special tours by reservation only (often you have to reserve well in advance, although occasionally you can luck out with next-day or same-day tickets).
St. Peter's Excavation (Scavi or Necropolis) Tour***
Part of the area under St. Peter's has been excavated to reveal ancient Roman burial sites and what is believed to be the tomb of Saint Peter himself. You must be in a guided tour with pre-arranged tickets to join—they often sell out well in advance, so plan ahead! See the Vatican Scavi website (although at least as of 2021 the website does not list their email address, which is firstname.lastname@example.org) for more information, as well as The Catholic Traveler's excellent advice on how to get tickets (also the St. Peter's Basilica website, in Italian).
The Pontifical North American College (a seminary for American priests) also can organize free tours of the Scavi (or arrange for an American seminarian tour guide if you've already booked the Scavi tour directly), more information here.
Vatican Garden Tour
You can book a tour of the Vatican Gardens on most days except Sundays; see the main Vatican Ticket Website for tickets
This is the pope's "summer home" which is now open for tours; see the main Vatican Ticket Website for tickets. In the past some tours included a train ride from the Vatican but they do not appear to be in operation as of 2021
Via Triumphalis Necropolis
This is somewhat similar to the Scavi tour but is a more recent excavation; ReidsItaly has good information about what it's like when it's open (there do not appear to be tours/tickets as of 2021)