I actually haven't listened to his symphonies...sadly
. I'm only slightly familiar with his string quartets (which, from what I've heard, are awesome!) Generally, I'm not too great on anything later than the romantic era (I should start listening to stuff, though)...
20th century is very diverse, way more than previous eras. Though it does have a lot of experimentation in sound, music theory, and the concept of music itself, there were many composers who wrote in a more "traditional" way, so don't dismiss 20th c. because there were a couple of Stockhausens in it. Check out Bartok, Poulenc, Vaughan Williams, Prokofiev, Britten, Schnittke (his late works), and Shostakovich.
If you're interested in symphonies - start with the 4th, then 5th, then 8th, then everything else (I'd recommend 6, 7, 10, 15 first). For string quartets seek records of Fitzwilliam or Emerson quartets, start with 7th and listen in ascending order 'til 15th. Also check out his cello concertos, violin voncertos, sonatas for violin and piano, viola and piano.
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