Let me start by saying that I know very little about classical music outside of the big names, and their major works. I'm not sure why I never took more time to learn about it.
Today, I caught a good friend sitting near a pond at my school listening to "Soave Sia Il Vento" from Mozart's opera "Cosi Fan Tutte." I wasn't going to disturb her, but she invited me to listen along with her. Honestly, I've never been more entranced by something (We listened to some more music, but I didn't dare break the silence to ask what each song was). After I finished classes I had work soon after, but when I got home I started combing the internet for classical music, and opera (I've actually become quite fond of opera pieces like "Habanera"). After maybe an hour of doing some reading and compiling an OK list of music I realized I hadn't even begun to scratch the surface. I'm here asking you wonderful human beings to give me literally all of the recommendations you possibly can, and maybe some resources on where to really learn more about the music. Anything you like, love, or feel is a necessity I want it. I think next semester I'm going to try to take a class focused on classical music or opera(Not to perform, but to learn the history and stuff).
EDIT: Ok so I have a 50 hour 17 minute long playlist. I'm gonna work my way through this over the next month or so. I want it to be a gradual process rather than just listening to one after another. I'm gonna report back in a month or so about my experience, the research I've done, and what I learned along the way.
You should look up a brief history of 'classical' music, because what is colloquially termed 'classical' includes a broad range of eras and styles. Mozart is true classical music (later, as opposed to Bach who pretty much started classical music, as opposed to early and Renaissance music, although it is really more of gradual shift). Beethoven on the other hand, the latest of the big three, is a Romantic composer. I think the best way to describe the difference is that Romantic pieces are 'showier.' Romanticism developed into various styles and substyles, which has continued to today.
If you want a good guide to the technical aspects, Aaron Copeland wrote a book titled How to Listen to Music. Included is one of the best guides to the essential pieces.
Here's a list of my personal recommendations as they come to mind:
1. Aaron Copeland -Appalachian Spring
2. Aaron Copeland- Lincoln Portrait
3. Dvorak-Symphony From the New World
4. Leonard Bernstein-Candid (whole opera, hilarious. On a side note, watch him conduct it. He doesn't really conduct, he just stands on the podium and dances. He's a genius.)
5. Shostakovich-Leningrad Symphony
6. Stravinsky- Firebird
7. Stravinsky-Rite of Spring
8. Mussorgsky-Pictures at an Exhibition
9. Handel-Water Music
10. Holst-The Planets
11. Vivaldi-The Seasons
12. Mussorgsky-Night on Bald Mountain
13. Gershwin-Symphony in Blue
14. Beethoven-Moonlight Sonata
15. Ives-Variations on America
16. Grieg-Peer Gynt
17. Sibelius-Finlandia
18. Tchaikovsky-1812 Overture (preferably with cannons)
19. Tchaikovsky-Swan Lake
20. Tchaikovsky-Sleeping Beauty
21. Tchaikovsky-The Nutcracker
22. Tchaikovsky-Symphony No. 6
23. Wagner- Entire Ring Cycle
24. Wagner-Siegfried Idyll
25. Schubert-Unfinished Symphony
26. Handel-Messiah
27. Stravinsky-Mass
28. Mozart-Requiem
29. Haydn-Symphony No. 94 (not a typo, he wrote a ton)
30. Debussy-Syrinx
31. Debussy-Claire de Lund
32. Debussy-La Mer
33. William Schuman-New England Triptych
34. All of Sousa's marches
35. Dukas-The Sorcerer's Apprentice
36. Strauss-Waltz Danube
37. Strauss-Also Sprach Zarathustra
I run out of steam there. The problem with remembering the amazing pieces is that they may have number names, and then it's just not going to happen for me. Obviously, I have a bias. Syrinx is only on there because I play flute. Russians are e extremely over-represented. I don't love the Big Three, and I detest Chopin, so they're pretty absent. But, hopefully based on some of these, you can find what you like. Given more time, I could triple this list pretty easily, but it's late, and bed is calling. Enjoy!