To be picky, Chopin, Stravinsky, Tchaikovsky, and later Beethoven works don't really qualify as classical. The middle two are Romantic composers, Beethoven began the Romantic era, and Chopin was perhaps the very first Impressionist, which is even further out than Romanticism. Alright. Rant mode off.
I rather like all composers. Shostakovich is very interesting, especially his String Quartet No. 8. Seriously some crazy stuff. Bartok I think was on crack when he composed. Rachmaninoff is just indescribable. Liszt, Ravel, Mendelssohn, Bach, Scarlatti, Handel, Haydn, Berlioz, Bizet, Respighi, Franck, Wagner, Mozart, Schubert, Khachaturian, Schumann, Orff, and a million other ones. Khachaturian is especially underrated. Go buy some.
Beethoven's Piano Concerto No 5 has been a favorite of mine for years. Bach's Piano Concerto as played by Glenn Gould is tremendous. Any Bach played by Glenn Gould is awesome. Sonata Pathetique by Beethoven is great. Liszt's La Campanella is fantastic. Damn, I should stop now before I really get started.
Peace.