A while back I was asked to provide some recommendations for people getting started in classical music, and I began by providing a "road-map" of sorts for your typical classical music section. And then, in my own indomitable fashion, I forgot about the whole thing.
Until now. [Insert rolling chords of portentous doom here]
The problem that comes up is mainly one of strategy. How should one enter the world of classical music? It's not that big a deal, really; there's nothing more daunting about making one's first real foray into classical music than there is about making one's first real foray into, say, poetry or sculpture or whatever else. But for some reason, a lot of times people take the wrong strategy for entrance, and thus classical music ends up overwhelming them. (Not unlike my high school's practice of tossing ninth-grade English students right into Shakespeare in the first two weeks of school. I've long been convinced that this practice more effectively kills any possible love of Shakespeare that might have otherwise developed.)
My own experience with classical music was fairly complicated: first of all, I loved film music long before I got into classical music in a big way, so I was at least passingly familiar with the ins-and-outs of an orchestra and the sound it makes. Also, my older sister was always into classical music as long as I can recall, so I was always hearing it in the next bedroom, even if I wasn't actively listening. And finally, I myself participated in the school band from fifth grade on, so I had some theoretical background in music before I got into classical in a big way. (This occurred roughly in ninth grade.)
So, I'm not sure how to present classical music to someone without similar background, someone whose exposure is significantly less than mine was at the time that I came to it. But for what it's worth, I tend to advocate not piling up collections of the "Old Warhorses" of the classical music literature in slapdash fashion, snatching up Beethoven and Bach and Mozart and Wagner just because they're, you know, the Old Warhorses. Instead I recommend that people begin their explorations with works heavy in both melody and drama and then branching out, often working backward. (And if you're really new to classical music, avoid any music described as "serial" or "atonal". Trust me. This kind of music is composed on a totally different set of assumptions from what you're used to, and you'll be right in the midst of "Shakespeare for thirteen year olds" again.)
So, I'd start with works like Rachmaninov's Symphonies and Piano Concertos; for the Symphonies, I would highly recommend Vladimir Achkenazy conducting the Concertgebouw Orchestra (which are helpfully available in a two-disc budget package, last time I checked). In fact, I'd recommend Ashkenazy in general with regards to Rachmaninov. With Rachmaninov, you get shimmering orchestrations, sweeping melodies, and a good healthy dose of brooding Slavic character. From Rachmaninov, one can then branch out into contemporary Russians: Scriabin and Prokofiev, definitely, and then Shostakovich.
Then, staying with Russians, go to Tchaikovsky and Rimsky-Korsakov. In the case of the latter, get a recording of Scherherazade. This is a spectacular symphonic suite inspired by The Arabian Nights, and its orchestration is guaranteed to amaze. Tchaikovsky is the greater of these two; here I'd recommend any (or all) of the Fourth, Fifth and Sixth symphonies, along with the Piano Concerto #1 (actually, this work is probably a bit overexposed, but it's a good way to find out what concertos are all about) and the ballets. Don't bother with the ballet suites, get the entire ballets. And if you must, get a recording of the 1812 Overture (make sure you get digital cannons), and the Capriccio Italien, although neither of these are favorites of mine.
That's probably a decent start for now. If you're like me in that your interest in classical music is fueled somewhat by an appreciation for film music, taking the "Russians First Approach" might be the best way to make the transition from film music (which tends to be all melody-and-drama) to music where form becomes more important. It's about training the ear to listen to the orchestra and to attend to longer forms of music than one is accustomed to, but with the emphasis on drama and melody typically in the Russian tradition, I think a newcomer might be a bit less "at sea" as they begin their explorations. Of course, one can't stay with Russians forever; sooner or later, a person exploring classical music must grapple with the Germanic tradition, which is by far the mightiest current in all of classical music. So that's where I'll try to go next time.