Advice above is good, especially concerning classical -- Beethoven 7 and Tchaikovsky 6 and that's it? Sheesh. Go buy yourself recordings of the Bach's Goldberg Variations, Brandeburg Concertos, and Violin Partita in D minor, Haydn's Cello Concertos, some Mozart string quintets, Beethoven Opus 110 and 111 piano sonatas and opus 131 string quartet, Berlioz Nuits d'Ete, Schubert's trout quintet, Chopin's Nocturns or Ballades, List B Minor Sonata, Brahms Violin Concerto and Sonatas for Cello and Piano, Mahler's 1st and 9th Symphonies, Debussy's Prelude to the Afternoon of a Faun, Ravel's Gaspard de la Nuit and Daphnis and Chloe Suite, Stravinsky's Rite of Spring and Song of the Nightingale, Bartok's Concerto for Orchestra and some of his string quartets, Prokofiev's 3rd piano concerto and 7th piano sonata, Barber's Cello Concerto and Shostakovich Symphonie #5. Then you will have a sampling of what are all themselves overplayed warhorses of the classical literature -- all of them great, none at all obscure. (Other people could just as easily generate equally valid and completely different short lists). If you get bored of these, then no worries, because each one suggests about a thousand other things to listen to. Oh, and you get the pleasure of being able to compare performances of the same pieces too. Gould's and Kirkpatrick's Goldberg variations could hardly be more different, and both are awesome things.
On the non-classical side, check out some Frank Zappa. He sounds like nothing else, and part of what's great about his work is that, while most of it is packaged in rouughly the format of standard rock band, his composing, with its myriad chord changes, quick time shifts, build and release crescendos, often extremely long melodic lines, virtuoisic passages and choral counterpoint, has virtues like those found in the best jazz and classical material, even of the hard-to-take-in avant garde varieties. So listening to Zappa with the right kind of ear can be a wedge into other music that is somewhat hard to grasp at first, but extremely rewarding when you learn how to hear it. I recommend "The Best Band You Never Heard in Your Life", and "We're Only in it for the Money" for easy entry into Zappa. Carefull though: When I started listening to Zappa -- long after being saturated in classical music for well over a decade -- I hardly listened to anything else for 3 years. Still not bored though.
As for jazz, get some Louis Armstrong and the Hot 5's and 7's from around 1927-1930. If you haven't listened to that, then you haven't heard jazz. Everone is still playing Louis' solos.
Man there's so much. No need to get bored.
RNM
RNM
