I'm extremely dubious regarding my ability to point anyone in the right direction for listening to classical music. I mention this as Rei asked me something along these lines the other day and I've been thinking about what I might usefully say since then. The problem is that not only am I no expert, I am so lacking in anything close to expertise in this area as to be painfully inadequate.
However, I suppose it is of some relevance and potential helpfulness that I've managed to go from being almost completely unable to apply myself to listening to such music to actually being able to apply myself to reasonably sustained appreciative listening. And it's of interest (to me, anyway) to note that this came about largely after a very intense experience in my middle-twenties.
I borrowed some recordings, old fashioned lps, mostly of the music of Ralph Vaughan Williams from a friend and suddenly, gloriously understood exactly the appeal of what I was listening to. The excitement of realising there was all this wonderful stuff waiting to be experienced was extraordinarily powerful, such that even at a remove of almost thirty years I can more than remember it, I can taste it, touch it. And once I'd 'got' VW so much else followed naturally, as it were, though perhaps not with quite the same intensity: Holst, Bax, Elgar, Britten (the more obvious Brits); Ravel, Debussy, Messiaen, Sibelius, Ives, Copeland, Gershwin, Bartok. And all these Moderns gave me a kind of access to the past.
So I guess I'd recommend listening pretty much to anything by VW as a starting point for someone looking for the same kind of breakthrough. But I have some reservations about this. Just because it worked for me, I'm not sure that would generally translate into instant understanding for others. In fact, VW seems to me so fundamentally wrapped up with a sense of Englishness that I'm very unsure he would work for anyone beyond those shores. But for anyone who's interested I'd suggest starting with The Lark Ascending and Fantasia on a Theme by Thomas Tallis. After that, you'd know.
Another composer who I seemed to 'get' effortlessly was Copeland. Appalachian Spring is a useful test piece. Again, if it works you're in.
In both cases, by the way, I made no real effort at the time to listen. I didn't need to. I think that was important to the experience, though I'm now keen to make sure I actively listen to any kind of music. That's oddly contradictory, but there it is.