You must remember that pieces in classical music weren't just made for the sake of just listening and enjoying the music. For some great composers, it meant more intellectual challenges and unheard melodies.
In fact, I would think of as enjoying classical music being the tiniest part of the experience. Its no brainer that some of the music won't get into you - the music was composed for a different style of audience and style, and what might be called masterpieces may be denounced easily by the public nowadays.
You really have to put yourself in the past for baroque and classical music, as some can get really hard to listen to. I can mention a couple of pieces that are really hard to listen to at first - the passions of bach, (just ANYTHING with Bach to be realistic) late string quartets of Beethoven, and for me, (contraversially(!)) everything by Mozart. The time effect is lessor in romantic music and contemporary.
So what does this mean? To get the true essence of classical music, you'll, if you don't your ears hurting, try to listen to more popular classical pieces at first, moonlight sonata, fur elise etc. (despite my disdain for it) Then move on.
Try this for a list:
Canon -> Moonlight sonata -> pathetique sonata -> appasionata sonata -> hammerklavier -> Bach solo cello suites -> Beethoven symphonies/ piano concertos -> Bach Violin partitas -> Everything by Mozart -> Everything by Bach.