Here is the Vatican Museums website, in English and Italian, plus a list of tips for visiting them. They are open for free on the last Sunday of each month. Get in line early, especially on the free Sunday: the lines are long and the museums are crowded.
A list of all the museums in Rome with hours and contact info: http://activitaly.it/musei
A general overview of all the museums of Rome, plus details on their various collections, but no office hours or contact info: www.roma2000.it/musei.htm
Museums run by the city of Rome, including minor ones: www.museiincomuneroma.it
Each year in the spring the Ministero per i Beni e le Attività Culturali has traditionally sponsored the "Settimana della Cultura," a week where all the museums are open for free. In the past few years this seems to have been replaced by the opening all national museums for free on the first Sunday of each month.
La Notte dei Musei is a night once a year when the museums are open for free, there are various concerts and exhibitions in the streets, and even the metro stays open late.
Rome's churches are a kind of free museum, and many of them have works of art comparable to those in museums. The Basilica di Santa Maria Sopra Minerva has works of Michelangelo, Bernini, Fra Angelico, and Filippo Lippi. Caravaggio's paintings can be found in the churches of San Luigi dei Francesi, Santa Maria del Popolo, and Sant'Agostino.
Archaeological and Historical Areas
For info on visiting the ruins of ancient Rome, see the official site of the Sopraintendenza di Beni Archeologici.
For more detailed history of these things, see Rome: Eyewitness Guide or The Blue Guide: Rome (10th ed.). The websites "Rome Art Lover" , "Roma Segreta" and “Roma SPQR” also have a wealth of information on history and art. So do Bill Thayer's Rome pages.
Many churches offer free concerts of high-quality classical, sacred and other choral music. Look for advertisements at the churches themselves.
Here is a place you can regularly find free concerts: The “Amici di Musici Sacra” organize free concerts in the Church of Sant’Ignazio, performed by a variety of international choirs and musical ensembles.
Every fall (October-November), the Fondazione Pro Musica e Arte Sacra organizes an International Festival of Sacred Music and Art. This involves a series of free concerts in the papal basilicas by the Vienna Philharmonic and other world-class artists.
InRomeNow lists concerts for the upcoming week, both free and paid, classical and modern.
www.romacheap.it is a site dedicated to free cultural events in Rome.
www.info.roma.it has information on current cultural events and art exhibits in Rome, along with many other useful links.
AngloInfo provides a good list of original-language cinemas.