You're one of the lucky ones, in a way. Way too many people have the opposite problem!
Somebody above mentioned that a lot of rock/jazz musicians seem to be more into music from composers like Ligeti, Bartok, etc... In a way, I'm somewhat part of that category and will freely admit that I initially learned a lot about classical music via really ludicrous avenues like Emerson, Lake, & Palmer and Yes (who used to open their sets with an excerpt from Stravinsky's
Firebird). As I mentioned elsewhere, a history teacher of mine also gave me a bunch of great Bartok and Stravinsky CDs. My bass guitar instructor (who was bizarre and rock-and-roll as hell!) even got me into studying bits from Bartok's
Mikrocosmos, which to my prog-rock/Nirvana-soaked ears (both of which I don't really listen to much anymore) was like opening the gates to a new world of ideas and musical architecture. I got into stuff like Liszt and Beethoven as well, but not quite as much as composers like Stravinsky and Bartok.
Being a bassist/guitarist, I would say that studying jazz theory did a lot to shape my musical trajectory. From the outset, you learn to get used to very thick modal harmonies and things like melodic-minor tonality, octatonic scales/harmonies, and dissonant alterations (b9, #11, etc..). After a lot of that, things like the Scriabin mystic-chord and the Tristan chord seem more commonplace and all varieties of chromaticism seem to make a sort of sense.
I'm planning on following the Boulanger approach and dedicating some serious time to studying Bach's music (especially the Well-Tempered Clavier and the suites) more closely. I've listened to those pieces a zillion times and I'm a huge fan of similar sets of preludes/fugues from the past century (Niels Viggo Bentzon, Walter Hus, Henry Martin), so I may as well become more well-grounded in Bach's original 48 pairs. As well, I really want to learn more about Scarlatti and Haydn, who I have next to zero knowledge about. Some stupid quiz I took on the internet about "what composer are you?" gave me Haydn as a result, and I felt like a dumbass for not knowing his work at all

As well, I've found myself feeling hugely inspired after I read a couple of items about Elliott Carter's first string quartet, one of my favorite chamber works from any time period. Prior to the composition of that masterpiece, Carter spent a ton of time pouring over the quartets of centuries past ("I read through all the Haydn, Mozart, Beethoven, Debussy, Ravel, Bartók, Berg, and Ruth Crawford Seeger quartets to find a way of using the four instruments to present my ideas."), as well as studying contemporaries like Nancarrow and Ives (whose influences are blatantly at work). Considering the fantastic results of drinking deep from all the different periods, I think that Carter's approach sets a fine example.