Okay, so I love Beethoven, he is one of my all time favorites. But I don't like Mozart and I think Bach is really cool. And that's basically all I know of the Classic music world. I want to know SO much more about it! And I'd love to hear about some good opera's too!
I think Mozart is a great entry into Opera, as it's relatively accessible -- but if not a Mozart fan, there's some good Verdi out there that's also accessible. As far as other things to listen to... I'm on a serious Shostakovich kick these days. There's a lot of exploring to do, from symphonies to chamber music. Although it sounds a lot less like what we normal think of from Shostakovich, his dance/jazz/ballet music is fantastic as well, and a huge contrast to his regular style. If you love Beethoven, you could even just stick with that and expand out from what you know -- listen to early versus late piano sonatas, quartets, etc. And beethoven does have an opera, Fidelio. Not performed all that often, but its overtures (Fidelio, Leonore Nos. 1-3) are famous.
I don't know much classical, but I have always liked Scheherazade. By Rimsky-Korsakov. Pretty, exciting, romantic, with a story behind it.
how can you not like mozart...you must not know too much. https://www.youtube.com/watch?v=ZC2ePGkmopg
Shoshtakovich is excellent, as is much of Russian classical music. Check out Mussorgsky's Pictures at an Exhibition and Prokofiev's Romeo and Juliet. Also listen to anything by Chopin--the high point of Romanticism (according to me). Personally, Bela Bartok is my favorite composer, but awfully thorny and hard to listen to for many people. Olivier Messiaen is another favorite, as is Stravinsky. Here's some more modern stuff to try: Stravinsky-The Rite of Spring, The Firebird, and the Symphonies Bartok-Concerto for Orchestra, Violin Concerto No. 2 Messiaen-Quartet for the End of Time Faure-Requiem Debussy-Children's Corner Suite
Take a listen,, Choral version of Agnus Dei sung to the theme of Samuel Barbers Adagio for strings.Performed by The Choir of Trinity College,Cambridge,UK. Amazingly beautiful https://www.youtube.com/watch?v=KkObnNQCMtM
See if you can find a copy of Classical Music by Phil G. Goulding. He gives his list of the 50 greatest classical composers, their life and times, and their best works. Plus chapters about the instruments, the makeup of a symphony orchestra, etc. A great reference.
Stravinsky is so fucking amazing, I'm listening to Rite of Spring right now. It boggles my mind that a man composed music like this 80 years ago. The use of dissonance is refreshing even today, and the work is filled to the brim with passion. I also recommend Dvorak's Symphony No. 9 (I think), AKA Symphony From the New World. Mozart's Requiem and Vivaldi's Four Seasons.
for some modern classical, I suggest the album Tabula Rasa by Arvo Part also, check out the album Rossz csillag alatt született by Venetian Snares - it combines modern classical with breakcore electronic music to create a very unique sound.
As many have said, Shostakovich. I love Shostakovich. Also, DEFINITELY look up Debussy and Mahler. Debussy is probably my favourite composer. Sibelius is pretty good, too. Also Dvořák and Stravinsky and Vivaldi. Ohh and I'm a Telemann fan. I don't have any specific pieces by any of them in mind, so just... kind of look them up generally.
Rachmaninov!! !!!! the 2nd and 3rd piano concertos are AMAZING. Mahler 3rd symphony is a great starting place Stravinsky the Firebird Suite is awesomeee. Rite of Spring is another good one.
Since everyone's so into Stravinsky, try to hunt down Kazuhito Yamashita's solo classical guitar arrangement of the Firebird. It's one of the more ridiculous things I've ever heard.
Niccolò Paganini Johann Strauss Antonio Vivaldi Giuseppe Verdi Sergey Prokofiev Schubert is pretty good too.