I could go on endlessly about some of the classical stuff I'm familiar with. In fact, since it's the end of the day and I have some time, I'll do that just now.
Part of my enjoyment of certain composers has to do with the circumstances under which each piece was written. One remarkable example is the Russian composer Dmitri Shostakovitch. For nearly his entire life, he was prodded and goaded by the Communist government to make music in a certain way, to the point where he actually feared for his life due to government response to some of his works. He bottomed out emotionally at around the time of his eighth string quartet, which he later described as a eulogy for his own funeral, having caved to pressure and joined the Party some time earlier. In that quartet, he continually returns to an ominous four-note theme that spells out his own initials "D.Sch" (the note names are different than they are in English). In one of the middle movements, the melody hammers percussively in a way that supposedly mimics the KGB banging on his apartment door in the dead of night. There's no happy ending to the quartet, no resolution, just the grim and somber return of the four-note theme.
Bela Bartok's string quartets are another source of enjoyment for me, and for similar reasons, though not as dour. For Bartok, his development as an artist was synonymous with one thing: moving away from traditional rhythm/harmony and embracing what was at the time considered "folk" music, whether it be Gypsy dance music or Hungarian bar songs, all of them in comparatively bizarre scales and meters compared to what anybody was doing in Western orchestral music at the time. He applied those influences in a unique way that stretched the boundaries of what four stringed instruments were capable of doing, because he really wanted to capture the untamed, almost mischievous element he found in his research of folk traditions. He was fond of inversions, playing a melody forwards, backwards, and upside down in strange counterpoint. One of his most well-known inventions was what later became known as the "Bartok pizzicato," whereby the player would intentionally pluck the string so hard that it would slap against the wood of the instrument with a loud snap. In some quartets, he developed a style he would return to throughout his life, the eerie sounds of "night music", with the instruments imitating bird calls, the faraway songs of farmers returning home at dusk, and a general mood of ambient uncertainty. And in his last quartet, the 6th, like Shostakovitch he responded to a political situation, this time the Nazi occupation. Deeply sarcastic, grotesque renderings of children's limericks mock the listener in one movement, and the lock-step march of Nazi soldiers is musically caricatured in another. But the overall mood is bleak by the end, as Bartok knew that this wouldn't be the first or last of such atrocities.
If string quartets aren't your thing, another distinctive and storied composer whose work resonates beyond just being an example of his period is Igor Stravinsky. His most famous piece is the ballet Le Sacre du Printemps, commonly translated into the English title The Rite of Spring. Rumor has it that small riots broke out in the concert halls where the ballet originally debuted. The whole thing is a celebration of pagan sex rituals, flesh sacrifice, and worship of the earth, startling subject matter from the devoutly Christian composer. It has some memorable moments, but is very 'cinematic' to me and thus a little one dimensional. My favorite piece of his is the Symphonies for Wind Instruments, possibly my favorite piece of music in general. I could sort of describe Bartok and Shostakovitch in words that mean something outside of the music, but here I just have to defer to the sound of the thing. Some of the savage and unpredictable atonalism of The Rite of Spring is on display, but confined to a collection of brass and woodwind instruments that carry me to a subtle and mystical realm every time I hear it. One bit of background is that the finale of the piece was originally a separate composition Stravinsky wrote when he learned of the death of Claude Debussy, a fellow composer (thought vastly different in style). As before, the debut of this work was met with confusion and actually laughter. As Stravinsky himself was conducting, he reportedly did not stop to explain that this was serious music and all that, he just turned and gave the audience a grin without halting a beat and kept going.
I guess my overall message here is: to get a sense of the piece and of the composer, find something whose sound you like and learn about the circumstances that inspired it. If what you find interests you, dig further into the same composer's related works to find other resonances and recurring motifs. Like authors and directors, classical composers usually have signature tropes they were known for, or stylistic approaches they kept returning to. For me, since the above composers were all active around the beginning or middle of the 20th century, that's where I tend to look if I want to hear something new, because things were moving in an exciting direction at the time with the advent of jazz and non-Western music.
