Congratulations, Kubla. You've stumbled onto some of the greatest music ever composed (in my opinion).
The 8th string quartet is actually a musical autobiography of Shostakovich's life. The four opening notes of the quartet (d, e-flat, c, b) comprise Shostakovitch's "musical signature," and is a dominant theme throughout the piece. The first time he heard the quartet played for him, Shostakovich wrote in his journal that the tears streamed down his face "like piss from a man from having drunk too much beer."
The second movement of the quartet was intended to represent the schizophrenic and tumultuous nature of Shostakovich's world under Stalin and the socialist regime in power at the time. As mentioned above, counter-revolutionary music was a big no-no at the time, and Shostakovich and Prokofiev were (at various times) on the hit list. So, Shostakovich had two catalogues of music going. One which went to Stalin and his censorship boys, and the other which stayed in his desk drawer. Another thing to listen for in the third movement... the two sets of three very abrupt chords are meant to represent the KGB knocking on the door.
I would also recommend that you try listening to the cello concerto. You will immediately recognize the theme of the first movement (it's identical to a passage in the third movement of the quartet, i believe). Anyway, it's a fantastic piece. The second movement is spine-tingling and erie! Rostropovich has recorded it several times (it was composed for him), but my favorite is his recording with Seiji Ozawa and the London Phil. Also on that disk is the Prokofiev Symphonie Concertant (also a phenomenal piece of music!). As a Yo-Yo fan, I would also recommend trying out his recording. It's coupled with an interesting recording of the 5th Symphony. It may just turn you back on to Yo-Yo!
The symphonies are also terrific. Heck, everything is terrific. I once played in an orchestra conducted by Shostakovich's son, Maxim. We played the 5th and 9th symphonies, and to hear Maxim talk about his father, his father's life, and his father's music is something I will always treasure.
Happy listening, Kubla. There's a lot of great music out there!
-Eric