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How about trying Saint Saens' piano concertos (5 of them). Rachmaninov's preludes. Faure's music is highly under-rated - you could look into his Nocturnes and Barcarolles, as well as some shorter works such as the Romances sans paroles. Granados Spanish Dances are quite charming. Lots of Brahms to look into as well, including the Intermezzos. Happy listening!
Also Tchaikovsky! Go to the Adult Beginner's Forum, there's a Tchaikovsky recital coming soon and a list of his piano works. I especially like March, June and October from 'The Seasons'. Also the violin concerto D major, and the 1812 overture.
John Field's Nocturnes are also very lovely (especially No.5 Bb major, No.10 E minor, No.12 G major).
Music written after 1925!! Hold off on the mid-century atonal and avant garde stuff until you've listened to more that's in between what you're used to. Like some late Debussy, Prokofiev, Satie, Shostakovich, and early Schoenberg will help you understand what's going on in later Schoenberg. And the need at the time to escape certain perceived limitations on music will help you understand why Cage and Berio felt the need to do what they did. Only after that try people like Boulez and Ferneyhough. Music from 1975 to the present is incredibly diverse. If you don't like one composer try someone else and it will probably be totally different in every way. Try Saariaho, Part, Gubaidulina, Birtwistle, Tabakova, Auerbach, Adams, Glass, Ades.
Any suggestions as to which pieces and which other composers I should start listening to?
Rather than suggesting specific pieces and composers, I would suggest listening to as many concerts on the BBC Radio 3 website as you can manage to find the time for. The range of music presented is broad, and it's a great way to get exposure to music you may not think to seek out otherwise.
Their "Composer of the Week" series can be useful, too.
They keep things online for a week, which makes it quite convenient to catch things.
By a lot I mean many of their works for piano (if not most) at least once, although I still haven't gone through Liszt's work thoroughly. I have a large collection and I generally listen to music through the day.