Frannie, that’s quite clever! And it reminds me of a recent revival of an obscure 18th century comic opera called The Dragon of Wantley, written by John Frederick Lampe, who IIRC was Handel’s bassoonist and who made a large heap of money off this little burlesque. (While looking up the composer’s name, which I couldn’t remember, I came across this poem, which may amuse you.) At the risk of spoiling the ending, the dragon is killed by a stout kick in the ass.
Back to the OP…
Prokofiev is mentioned briefly above, and I’ll just mention Alexander Nevsky and Romeo and Juliet (not as mushy as Tchaikovsky’s version) as particular works you may enjoy.
Beethoven, Brahms, and Mozart all wrote good symphonies, but you knew that already. Other orchestral stuff not previously mentioned: for something old, try symphonies by Johann Stamitz; for something new, try Cantus Arcticus by Einojuhani Rautavaara.
For operas to start out with, you can’t go too far wrong with Mozart’s late operas (Don Giovanni, The Marriage of Figaro, and The Magic Flute) or Rossini’s comic output (The Barber of Seville, The Journey to Rheims, etc.). I’m partial to Handel operas – Semele, Julius Caesar, Xerxes – but I realize that these can be boring if you’re not really into 18th century music.
I love organ music (Bach, Franck, even Messaien), but only performed live. IMHO organ music has a strong visceral component; if you can’t feel the 16-foot pipes vibrating your innards, you’re missing a big part of the experience, and you just can’t get that from recordings. I’d sit in the pipe ranks if they’d let me. 
Choral music: I’m always happy to push Edward Elgar’s The Dream of Gerontius and William Walton’s Belshazzar’s Feast as choral works that needs more airing, especially outside of England. Ditto Vaughan Williams’ Sea Symphony (which, despite the name, is also a choral work of the lush, sweeping variety). Benjamin Britten’s War Requiem is a bit more challenging for someone just starting to venture outside the “pop classical” pieces, but it’s worth knowing – the interpolation of Wilfred Owens’s poetry in the Requiem Mass text is intensely moving. I also recommend a lesser-known Britten war-related work, Ballad of Heroes, although I don’t know what recordings of that are available.
Chamber music: My tastes (and recommendations) here are pretty eclectic: try Messiaen’s Quartet for the End of Time, Arthur Foote’s Night Piece for string quartet and flute, Schubert’s String Quintet in C Major, Beethoven’s Suites for Cello and Piano, and of course the Bach Cello Suites (recorded by just about everybody, and never the same way twice).
I think that’s enough for now.
(Oh, and it’s “Hildegard von Bingen”. I recommend the “Feather on the Breath of God” recording.)
