FolsomBlues:
In my opinion the worse thing you can do is get a compilation album. Classical music asks things of the listener (concentration for 30 mins and upwards) that popular music doesn't. I always think its a shame when people plug for compilations that has excerpts from slow movements (adagios) divorced from their context. I think its better just to dive in there and start having a proper listen.
Give Parsifal a go. Listen to it a few times without reading the libretto just to get a feel for it.
Don't be intimidated thats the main thing and don't always go for the obvious. Beethoven's ninth, as Stravinsky points out, is very boring in parts. Talking of Stravinsky The Rites of Spring is a good starting point for 20th century classical music.
"classical music" is an 18th century phenomena but we don't need to get into that.
Varese is someone who is easy to like from the 20th century. Its not all eine kleine bl****** nachtmusik and Rach 2s.
Stay away from boring tedious gimmicks like Asian violinists in short skirts and the Classic FM brigade.
Haven't laughed so much in a while! I agree, don't go for compilation albums. I would say that Parsifal may a bit much for a newbie! (Indeed, I tend to find opera, particularly Wagner, works much better on DVD).
I would suggest listening to some full symphonies, Beethoven would be my starting point as all of them are good works, and at least 5 of them (1,3,5,6,7) being truly great works imho. Mahler & Nielsen are two more symphonists I have a lot of time for.
Stravinsky ballets are great. Rite of Spring is fantastic, but the Firebird and Petrouchka are more approachable.
But to be honest, I'd just pick stuff up, give it a listen, and see whether it floats your boat. Your local library can be a good source of core repertoire, and you have little/no risk as it costs maybe a £1 to borrow a CD.