If you're new to classical, the first place you ought to go is the "overplayed" stuff. Chances are you've heard it way less than "I Want it That Way."
Here are some uber-popular favorites from the well-known composers.
Bach: Air for the G String
Beethoven: Symphony # 7 (a french horn playing friend of mine loves some of the prominent high notes from the horn in the 1st and 4th movements).
Bizet: The opera "Carmen"
Brahms: Intermezzo in c# minor op 117 no. 3
Chopin: Basically any nocturne or mazurka, but especially nocturnes. Op 27 no. 2 in D flat major is superb
Dvorak: His New World Symphony (#9) is most obvious choice, but he's also famous for cello concerto and Slavonic Dances
Hadyn: His trumpet concerto is rather fantasic, plus he wrote 104 symphonies ^^
Liszt: Les Préludes
Mahler: Symphony #5, 4th movment "adagietto"
Carl Orff: "O Fortuna" from Carmina Burana. This one is used in commercials a lot.
Puccini: "Nessun Dorma" from Turandot. It's basically the most famous opera aria of all time.
Rachmaninoff: The. Third. Piano Concerto in d minor. Also known as "Rach 3" (pronounced "roc 3")
Johann Strauss Jr: The Beautiful Blue Danube waltz. You probably heard this many times before, and in "Titannic."
Richard Strauss: Also Sprach Zarathustra, you already know this too, but probably didn't know its composer. He also wrote two of the raciest operas of all time, Elektra and Salome.
Tchaikovsky: Sixth Symphony in b minor. The guy died 9 days after its premiere :/ He even claimed it his best work. But it also despairs, be warned.
Verdi: "Requiem"
Wagner: "Prelude and Liebestod" It's a condensed version of his opera "Tristan and Isolde." Basically it's the overture and finale of the opera, but the finale part will be instrumental instead of sung like in the opera proper. It's an easier way to get into a four hour opera :D, since frankly that can be rather exhausting. But lemme tell you when you get to that final scene after listening to the whole opera, it's that much more satisfying. I cried totally unexpectedly.
No, I didn't forget Mozart, but I'd go on and on forever anyways and so I save the favorite for last. If you are worried you won't like opera, begin with "The Magic Flute." I began there, and though I've "branched out" to other opera composers, this is my favorite of any music. His symphony #41 has a really nice posting on youtube concerning the 4th movement, in which the person who uploaded the video talks about the 5 part fugue near the end. Oh and I've never tried posting a youtube video...we'll see what happens.
http://www.youtube.com/watch?v=Fcly8-RGhgwSome other favorites that fall under "Pledge right now and we'll end this drive"--the music always played at NPR pledge drives: Barber's Adagio, Elgar's Nimrod, Rodrigo's Concierto de Aranjuez, Handel's Royal Fireworks, and the list keeps going. Oh yea, Debussy <3 Claire De Lune.