Having gone through that process of switching from genres such as pop and rap (which now are too boring and repetitive to me) to classical, I'm pretty well suited to answer the question. Over the era of classical music, music progressively became more complex and difficult for untrained ears, so I started at the very begging, with baroque. This means composers like Vivaldi, Pachelbel, Telemann and especially Bach. That will introduce you to the Classical period, with composers like Mozart, Schubert, Haydn, and others. In the romantic era, things start to get a bit intense with Beethoven (his later works), Paganini, Sibelius, Tchaikovsky, which then lead into the incredibly complex period of neo-classical/20th Century/contemporary. These are composers like Shostakovitch, Prokofiev, Schoenberg, Stravinsky, etc.
Feel free to listen to all of these eras as much as you want, (this is not strict), but that's probably the musical path you'll take.
So, some pieces I'd recommend to start with:
Vivaldi's Season's (a collection of 4 violin concertos) and others by him
Some of the Beethoven symphonies are quite nice to start with
Bach - Preludes and Fugues from the well-tempered clavier (my favorite is no.2), and others
Well, these are the 3 i can think of for starters, but it'll be a long process.
Later works just to see what you're in for
Sibelius Violin Concerto, Shostakovitch Violin Concerto, Prokofiev Flute Sonata, Stravinsky's Firebird.
So, figure it out!
For the last question - There are many excellent performers today (e.g. Yuja Wang), but not as many amazing composers. But you don't have to look to far back. Many neo-classical composers wrote beautifully for piano.