interesting...so let's see....
i would think that playing is the most fundamental. i would use any devices you can some up with to push you into it. find a practice room.
basically improvising is like composing--it's one way to go about it--so one of the tasks that awaits you is shifting how you think about structure--from elements you repeat (from the score) to elements you organize yourself.
so for example: if you have a short segment of a piece that you like, try isolating it from its context and playing around with it. when i was a kid, i used to have great fun taking apart piano rags. but you can use anything, really. one little game that i liked was adding stuff between sections so that i had a place to start from and a place to get to. it was like a puzzle.
another fun game--but its time consuming--is learning a classical piece backward.
you can use harmonies that you make---try making a series of ninths with your left hand, move them around chromatically and see what you can do with your right to fit a melodic-ish line over it.
ninths are pretty. you can augment or diminsh them and the fact that the pitches are spread out a bit will erase the effect of the dissonance.
use graphic elements---i used to like using kandinsky paintings as a kind of score.
stuff like this:
there's lots of ways to interact with these shapes--literally transposing some of them is one option. i had a poster when i was a kid that was more horizontally organized. i used it alot as a kind of score. it helped to have a visual referencepoint when i was figuring out how to use variation. that was one of the biggest shifts--getting away from using chord changes to organize the sound, trying to do more counterpoint based stuff.
another game (maybe for later, but you never know with these things) was: i got really interested in ornette coleman's music. there's no piano in alot of it...no chords...just multiple lines. i spent a LONG time trying to figure out how to do walking bass and what i thought of as horn lines simultaneously.
i have made quite a few decks of cards with different graphic or verbal cues..shuffle and select a few.
play that.
there are lots of tricks/games/things you can experiment with.
so i would start playing.
listening:
as for schoenberg/webern/berg: at some point you may listen to pieces again and find that you hear them quite differently than you do now. webern's op. 30 for piano was for me at least fundamental. maurizio pollini has a really lovely recording of it on dg, doubled up with an early boulez sonata that is just great (and introduces you to "motivic cells" as an organizational tool) but i suspect that it'll take you a while to work your way into it. i'd rip it and keep it around, play it from time to time. you may find that one time you play it, you no longer think "how strange" and just drift into the piece. that's when you are starting to hear it.
i would check out the webern, though. there's a world in it.
here's a couple more recommendations, based more on what you've posed since the op...
satie--try to find steffan schleiermacher's recording of "vexations." it'll mess with your head.
later debussy (around world war 1) and scriabin (tho scriabin is much busier) were both fascinated with the wonderful world of fourths. those are nifty intervals to play around with.
keith jarret. there's alot of nice stuff, especially from the late 1970s---the american quartet (with dewey redman, charlie haden and paul motian) recordings are all great; there's a very pretty orchestral record called "arbor zena" that is well worth tracking down. people like the koln concerts, but i never did. it's mostly pretty straightforward arpeggio based riffwork. but folk like it, so maybe you will.
bill evans.
lennie tristano.
you might also like some stuff by any of these folk:
warne marsh
art pepper
jimmy guiffre.
who are all horn players but did most lovely stuff.
eventually, check out ornette coleman. his first release, "this is our music", is the gateway drug.
it's like milk is for junkies.
all junkies have consumed milk.
it must be the gateway.