The only wise way to go, I suppose, is to listen. And listen, and listen.
When I was in your age I played a little of Beethoven and to tell the truth, I hated it. What a boring old man ... OK, so I enjoyed playing the first movement of the Moonlight Sonata, but the rest -
Today, I cannot understand my younger me, because I am totally in love with Beethoven. He was an excellent pianist, an excellent composer and he laid the whole foundation to modern rock music, IMO. (Of course not the only one!) And he is always fun to play, there is always something new to discover! So today I'm a Beethoven junkie, always in the need of my daily fix.
So a start could be to listen through all his 32 sonatas, because they are a great journey through a great composer's life. When he wrote the first ones, he was young and a big admirer of Mozart, and also a student of Haydn. You can hear that. Later on he developed his own style and also took advantage of his amazing improvisational skills. He lived his life, he fell in love, had his heart broken and later on he had to struggle against his developing deafness. These passions and torments are all reflected, somehow, in his music. When he wrote his last sonatas he was totally deaf. The last movement of the last sonata is totally amazing, as it is partly a boogie woogie piece ...
But, take it easy. Going through all the Beethoven sonatas is not something you do in one evening or two, especially not if you are new to this area.
And, speaking of Beethoven: Für Elise is a must for every piano student.
If you want something really romantical, I suggest you check out Schubert and his lovely Improptus.
And, considering the pieces you mention you have played, I wonder if not Debussy and his impressionistic music is your cup of tea? Try Clair de Lune, Rêverie, The Girl With Flaxen Hair, the two Arabesques. Beautiful, dreamy and interesting music, I love to play Debussy.
Topic: Introduction to classical pieces?