Educate me. Help me understand and enjoy it, or at least appreciate the concept better. I find it an interesting formal experiment, but not really. . . enjoyable? For example, I’ve been listening to string quartets no 2 op 10; no 3 op 30; and no 4 op 37, if a point of comparison/pointing me to examples of something helps (if you can suggest other stuff to try that would be good, too)
(I consider this fair trade for spending too much time in modern art threads)
I’m not a musicologist; I also don’t think a musicologist would necessarily help, except to try to give you intellectual reasons to appreciate Schoenberg. I’m a performer and a fan, and I’d like to see if I can help you out on any kind of level.
I find the Schoenberg’s music is incredibly sensitive to performance - if the performers approach it as abstract, austere music, it comes out that way. There is one massive point to never miss in Schoenberg’s music - he believed passionately that he was the latest link in a chain that went through Bach to Beethoven to Wagner to Richard Strauss to Mahler to Arnold Schoenberg. He believed that he was the reunification of the music of Brahms and the music of Wagner and Wagner’s followers. Once, late in life, someone asked him why he had not written more music like “Verklärte Nacht”. He apparently took off his glasses, pinched his nose and with the pained expression of one who is perpetually misunderstood, said “I have been writing nothing but music like Verklärte Nacht”.
So, I’d encourage you to try to follow the through line of AS’s writing. Starting with Guerrelieder and Verklärte Nacht, then listening to Pierrot Lunaire, the 5 Orchestral Pieces (Farben is just heaven if it’s played right.) and the Chamber Symphony, then listening to some of the later things like the 4th String Quartet or the Ode to Napoleon. Go back and forth, and listen to how often he liked to waltz, and how his music should have the same lilt as Johann Strauss, just with chords that Strauss would never have considered consonant.
I don’t know if playing through any of his stuff is an option for you (I’ve sung some of his songs, but his piano music is way beyond my present ability), but if it is, even just a couple of bars will change how you think of his music. The piano music is a good place to work from, as you can hear the changes in his style.
But above all, if a performance is making you think of words like harsh, abstract, anchorless, then I think the performers are on the wrong track. If a performance is making you think of romanticism bordering on neurosis, that’s closer to what he was after.
Absolutely.
I’d strongly recommend also plunging into the world of Webern. Try the string quartet op. 28. An advantage is that it’s typically brief in length, less than ten minutes total, and so it’s straightforward to quickly become familiar with it on a certain level through multiple listenings.
However, he didn’t write miniatures. Every gesture, even down to the level of single notes, carries a great burden of intensity. The self-imposed restrictions of serialism force all other aspects of the sound world (a wide dynamic range, the characteristics of individual sounds, and don’t forget about silence, too) to connect with the listener in a more immediate way than they would or could if pitch and harmony were the driving features.
In short, don’t listen to it as ‘a serial piece’, but as a tightly-woven intricate world in its own right.
I take offence at that 
Edit: Glenn Gould’s Webern recordings are another great starting-point
Great answers. Thanks! I’ll give Webern a shot, too.
Just to round out the Big Three of the Second Viennese School, I’ll throw in a recommendation for Berg’s delightful Chamber Concerto (or Kammerkonzert).
Of Schoenberg’s 12-tone pieces, I really love the Suite, op. 29. Wonderfully colorful piece, scored for three clarinets, three strings, and piano.
GorillaMan Many apologies if I gave any offence - I’ve just encountered one too many musicologists who want to turn 12 tone into crossword-puzzle music. I remember a dress rehearsal for the Webern string trio, Op. 20 and some bugger did a chat before the piece. He was going on and on about recursive form patterning and set theory and Shenkerian analysis and God knows what all when the Glaswegian who taught viola that year yelled out “For Christ’s sakes, get on with it - you’ve havered on longer than the fuckin’ piece is gonnae be.” It was more or less what all of us were thinking, but I don’t think anyone else would have said it, nor said it quite so colourfully.
I like musicologists, though - really, I do… 
Biffy the Elephant Shrew - A lovely piece, but may I recommend the Violin Concerto as a stunning example of 12-tone coming out like Mahler? Or the Lyric Suite - if the performance doesn’t make you feel kinda randy, they missed the point. There’s also the Lulu suite (or Lulu, if it comes to that; I don’t think it’s a great work over all - I just don’t think Berg understood the bizarre Feydeau farce that is the 2nd act, and the Paris scene is God-awful, but if you can listen to the Prologue, Act I and then skip straight to Act III, sc. 2 after having read the play or watched the movie, it’s a life-changing experience.) Berg and Webern also have really cool through lines, esp. if you listen to the early Berg songs and jump to Wozzeck and then to his 12 tone stuff.
A really fun concert I’d like to try is to do some Dufay or Josquin and a couple of Bach WTC Preludes and Fugues for the first half, and then one of the Webern cantatas for the second half. All of the pieces have the same intricate formal basis, and all of them are beautiful even if you don’t know isometrics from calasthenics… Just IMHO.
I unfortunately know all too well the type of musicologist you describe, although I think they would be more meaningfully described as a ‘musical technician’, perhaps. I also have both musicologist and composer friends from Glasgow who would have offered similar contributions to the one you recount 
Now THAT is sounding like fun 
It’s not far beyond the kind of programming these guys have been going for (very highly recommended), but would take same serious logistics to get the right venue and players!
But I bet one could approximate it with an itunes playlist. . . On it!
I heard the following today -
A twelve-tone composer named Hall
Has a dodecahedrical ball
And the sum of its weight
Plus his pecker’s, times eight
Is his phone number - give him a call!
Who says musicologists aren’t good for a laugh?
What’s the definition of twelve-tone music?
Six desks of violas.
Le Ministre de l’au-delà,
I can’t tell you how to format indents for poetry, but can assure you that your problems are due to what I think of as standard web-based formatting–which mostly ignores spaces-- rather than specific to your browser. I believe it can be done, it just takes someone cleverer than I to do so.
Use a code tag.
A twelve-tone composer named Hall
Has a dodecahedrical ball
And the sum of its weight
Plus his pecker's, times eight
Is his phone number - give him a call!
-FrL-
Well, if we’re going to devolve into that…
What’s the difference between the first desk of violas and the second desk of violas?
About a measure and a half.
(Of course, now we’re into Ives, which is another discussion entirely).