The way I stick my toe into new music is to pick an achievable slice of music that interests me. I started with the Russians (Tchaikovsky, Mussorgsky, Rimsky Korsakov). I looked at the Penguin Guide and figured out five major pieces to start with. I used the guide to find good solid performances on one of the major label's budget line. No need to pay full price in classical music. There's tons of great performances in great sound in the back catalog. When I got the CDs, I chose one and sat down and read the liner notes, and listened carefully, following along with the track list so I knew where I was in the music. I had a few basic classical books, but the Internet would be better now. As I read up on the pieces and the composers, I'd read references to other pieces and composers. I kept a list and went back to the Penguin Guide and picked a few more, branching out from where I started. Once I felt like I had a little grasp of that area, I chose another and did the same thing... Impressionistic French composers like Debussy and Ravel, bombastic Romantics like Wagner and Bruckner, early Modernism like Stravinsky, etc... I followed the breadcrumbs and traced the branches as my interest carried me. If something didn't immediately hit me, I set it aside and usually, later on I would have more of a frame of reference for it. (I did that with Mahler. The structure was so broad I didn't hear it at first.) I use the same approach to learning about other kinds of music too, and have found incredible treasures... Hawaiian slack key guitar, western swing, Harlem jazz, pop vocalists, Cuban mambo, Kentucky folk, post Bop, Honky Tonk, bluegrass, Balinese gamelan, etc.
if you'd like to choose a starting point, I'd be happy to suggest five great CDs for you. That's the other way to learn... Find old record collectors and pick their brain. I hang out with musicians and musicologists all the time and I have more great leads to check out than I can ever follow up on. I've found that whenever vie gotten into a slump where music sounds boring, it isn't the music's fault... I'm boring. Hope this helps.