I suggest the following (with examples):
Chopin: Start with the Mazurkas and Nocturnes
Mazurka in b minor, Op. 33, No. 4
Nocturne Op. 9, No. 2 (you may already be familiar with this one)
Schubert: Winterreise song cycle
Der Lindenbaum
Shostakovich: What clutnuckle suggested, and also the Jazz Suite
Jazz Suite No. 2, VI. Waltz No. 2 (this is easily the most famous movement)
His string quartets are also excellent
Mozart: to get into Mozart, I always suggest listening to his Sonatas and Fantasias (Fantasies) for solo piano, and I also suggest his Requiem.
Lacrimosa from Requiem
Piano Sonata in C Major, K. 545, and Fantasia No. 3 in d minor, K. 397
Brahms: Brahms is so extensive, but this is one of my personal favourites and a good one to start with:
Symphony No. 4 in e minor, Op. 98, I.
Mendelssohn: His violin concertos are beautiful. Also his Wedding March, which you probably know already.
Tchaikovsky: Of course there are his ballets
The Nutcracker and
Swan Lake, but he has other works too! I suggest starting with the 1812 Overture.
Unfortunately, I'm not as familiar with the works of Borodin or Dvorak, but if you like what I've posted here I can recommend you other, similar composers
