Here is a list of pieces for you to start with if you want to get into classical music, but are overwhelmed by the sheer volume of composers and pieces available.
If you're just starting out there is no point to listening to every single Bach organ work, you should just be getting a feel for what you do and don't like. This list will help you do exactly that, without getting bogged down in the details. For each of the major eras in classical music, from Baroque to Contemporary, I've listed one or two pieces which really represent that period.
Everybody likes different styles. For example, it took me about two years of listening before I could bear to listen to the late romantic stuff. If you listen to everything in this list you'll have a really good idea which styles get the blood flowing, and which make you yawn, and then you can expand from there.
Bach is the superstar of the Baroque period, and the Brandenburg Concertos are one of his most well known, and most respected pieces. They are a varied bunch of relatively short (compared to the more elaborate and lengthy concertos in later eras) but perfectly formed little beasties. They demonstrate the technical precision and cleverness typical of Bach, and Baroque music in general.
These are the last two symphonies of Mozart, and show off his always perky, pretty, and elegant melodies. They are catchy too, see if you can avoid whistling the final notes of these after hearing them a couple of times.
Choose the 5th for a heavier, more in your face experience. You'll have certainly heard the first few bars about five thousand times already, but the rest of it (especially the third and fourth movements) are golden. It's a classic example of Beethoven's ability to mesh delicate, introspective music (3rd movement) with boistrous, glorious, triumphant excess (4th movement). On the other hand, if you'd prefer listening to something totally unfamiliar, symphony number 7 is lighter, more fun, much more rhythmic, and chances are you haven't heard any of it before.
The Mendelssohn piece is a reasonably conservative but fantastic piece. It's got bounding, driving rhythms, beautiful orchestration and melodies, and a cyclic ending that finishes with a transformed version of the start of the whole thing. The latter is wild, crazy, over the top romantic piano at it's best. you might know the rhapsody No. 2 from the Tom and Jerry cartoon "<em>Cat Concerto</em>"
Tchaikovsky is exceedingly sweeping and heart-stringy. Brahms is a little more serious, cultured, grand and imposing, especially the excellent last movement. Dvorak is more rhythmic, more jazzy, more modern sounding. All of them are big-R Romantic: big, emotional, expressive.
The atonalists (Schoenberg, Berg, Webern) didn't believe in scales and keys (Like A minor, B flat, etc.), and they gave every note equal importance. Unsurprisingly this makes a lot of their stuff very hard to enjoy, unless you are being all scholarly (or pretentious) about it. Although it does sound very interesting. This piece, however, is a unification of their techniques with "regular" tonal composition. It's a painfully emotional effect, especially if you read about the circumstances it was composed under.
These are all great examples of how many modern composers pulled back a bit from the extremes of the atonalists. They kept tonality, but pushed at its boundaries. Shostakovich uses melodies which are morbidly stuck between keys, Prokofiev makes it sound like someone is hitting the wrong notes (but, you know, in a good way) and Stravinsky caused riots with the primeval rhythms of the "Rite of Spring". The first is big, touching, driving, sarcastic and sly. The second is similar but more percussive, and in places more playful. The third is wild and syncopated.
And he we are today. This is music composed in the last couple of decades, and it is very far from the stereotypical ideas of classical music. The former is nothing like a traditional chamber symphony: it's a self-described marriage of atonal music with Looney Tunes cartoons. The second is one of Schnittke's many attempts at unifying "low" and "high" music. He'll switch from very classical, to ridiculous fairground music, to Psycho style stabs over just a handful of bars, while being horribly, wonderfully, screechingly microtonal (playing the spaces in between "regular" notes).