I would certainly caution anyone new to 20th-century classical music not to be snared by the trap of thinking the avant-garde to be the indicative, dominant, or best strand(s) of music. There's a vast array of 20th-century music that's more readily approachable and conventionally beautiful. I don't imply that the latter is superior, but I do recognize that's what most listeners want, and it would be a crying shame were they to miss many riches of 20th-century (and contemporary) music because of misguided fears.
And I would caution anyone new to 20th century classical music not to be snared by the trap of thinking that "the avant-garde" (to borrow Grazioso's misleading term) is less approachable or less beautiful than some of the more conservative (regressive) strands. It
is a crying shame that the persistent narrative that Schoenberg, Varese, Cage, Stockhausen, Xenakis, Dhomont, Boulez, et. al. are "difficult" or "inaccessible" or "unmusical" (and other canards) does indeed put people off a lot of very fine music before they've even heard it, and continues to keep them from enjoying it even after they've heard it. (I was at a concert recently that included Schoenberg's third string quartet, a lovely piece with easily recognizable themes. The first comment I heard, from a guy in his thirties I would guess, was that this is music that's really more suited for analysis than enjoyment. He cannot have heard the ravishing sounds those four musicians had just made.)
Indeed, I would caution anyone new to 20th century classical music not be cautious.
Otherwise, I do wish our discussions could get out of the tonality/atonality pit. God, there's so much more to 20th century music than that distinction! There are rhythmic explorations, instrumental explorations, at least three separate explorations of "noise," the most recent called simply "noise music." There is indeterminacy (giving up control) and aleatory (giving up some control in carefully controlled ways) and multimedia and mixed media and danger music and anti-music and Fluxus and minimalism (which itself has several distinct strands--think Young, Reich, and Feldman, at least) and polystylism and spectralism and electroacoustic and live electronics and improvisation and concept music and soundscape music and computer music and turntablism and what I like to call "tafelmusik," music created by playing miscellaneous items--acoustic, electric, electronic--spread out on a table.
In most of those, pitch is not the primary component (as it is in tonal and serial musics). Indeed, I often wonder if anyone talking about the contrasts between tonal and "atonal" (a mostly meaningless term) and serial realizes how close these are--they all use relationships between pitches to construct musical compositions.
That's my sermon, and I'm stickin' to it!!