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). Among the best-rated tour groups are Eden Walks and Angel Tours; TravelAdvisor lists other recommended tour groups. Open M-Sat 9-6, last ticket sold at 4. Open and free (and thus crowded) on last Sun of the month, 9-2.
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). The Spiritual Destinations website page on St. Peter's is incredibly thorough and has some great photography. Also see the very useful Saintpetersbasilica.org Travel Information page.
Pope's Schedule for a full list.
Santa Susanna Church, the home of the American Catholic Church in Rome. They offer an English mass every day at 6 PM at their church on Via 20 Settembre, near the Piazza della Repubblica. Fr. Greg, the pastor of Santa Susanna, leads occasional religious walks for English-speaking Catholics in Rome; his church also leads a pilgrimage prayer on Monday mornings. Their website also lists convents that offer cheap rooms for pilgrims and a list of English-language church services in Rome (for various denominations, not just Catholic).
The Vatican offers a few special tours by reservation only.
Every day at 10 AM, except Wednesdays and Sundays, you can see the Vatican Gardens. See the US Embassy website for more information.