FYI, there's (usually) 4 movements to a symphony, 3 movements to a sonata, and 3 movements to a concerto. If you start downloading (say) a symphony, see if you can tell the conductor/label/something identifying it, and go to Amazon and look up the CD. Then make sure that you have all of the tracks. It'd suck to listen to the movements of a symphony out of order, or to have (say) the 3rd one missing.
Beethoven - Symphony No. 5!!! Get this and listen to it continually until you memorize the whole thing! (NOT just the first movement!) It's probably the most revolutionary and influential piece of music ever written. Ever.
Prokofiev - Romeo and Juliet (very long piece (2 CDs) most famous part is traditionally called "Capulets and Montagues")
Tchaikovsky - 4th, 5th, and 6th Symphonies (6th symph. MUST be listened to in its entirety!)
Van Cliburn playing Tchaikovsky's Piano Concerto #2 (I think it's #2) (this is the best-selling classical music recording of all time)
Mozart - Symphony No. 40
Chopin -- waltzes, nocturnes, ... (lots of shorter pieces -- all for solo piano -- one of the sonatas has "Funeral March" or Marche Funebré as its second movement, which is really famous and quite beautiful when not being played by some Soviet marching band upon the death of the Premier)
Beethoven - Piano Sonatas. "Moonlight" is very famous. My favorite is "Pathetique".
Mozart - Requiem (it's in like 9 pieces)
Dvorak - Symphony No. 9 "From the New World"
Smetana - My Fatherland - (famous excerpt: "The Moldau")
Pachelbel - Canon (very famous piece)
Bach - The Art of the Fugue (many pieces to this)
Bach - Toccata and Fugue in.. hmm I forget the key, it's his most famous organ work though. Just type "Bach Toccata" and you'll get it.
Bizet - Carmen (excerpt: Toreador Song)
Barber - Adagio for Strings
Tchaikovsky - The Nutcracker, Romeo and Juliet Overture
Albinoni - Adagio (there's many of his adagios, but there's one that's really famous, it's in G minor)
Rossini - William Tell Overture
Rossini - The Barber of Seville (Overture, "Figaro", ...)
Debussy - Claire de Lune, Le Cathedrale Engloutie (sp?), Children's Corner
Händel - Water Music, Music for the Royal Fireworks
Elgar - Pomp and Circumstance Marches (You know the graduation song..... but you only know the suckiest part of the piece!)
Elgar - Enigma Variations (there's like 15 -- the most famous, and best, is "Nimrod" -- but try and get all of them and listen to them in sequence)
To introduce you to a bit of opera, I would recommend that you get the songs off of the CD "The Three Tenors in Concert", the London recording. Most of the good stuff is from Puccini operas.
edit: other people already said them, but Holst's The Planets and Mussorgsky's Pictures at an Exhibition are both really good and you've probably heard them before somewhere.