In case you're not familiar with classical music, don't be intimidated and give some of these pieces a try! The pieces I listen to are certainly more accessible than most experimental rock or jazz (yet those genres are way more popular). I think the symphonies and concertos are a good introduction. If you like intense music, maybe start with Mendelssohn, Tchaikovsky, or Beethoven. My favourite composer is Brahms.

Symphonies

• Beethoven No. 5 (Kleiber)
• Beethoven No. 9 (Barenboim)
• Brahms No. 3 (Szell)
• Brahms No. 4 (Bernstein) (Kleiber)
• Tchaikovsky No. 5 (Karajan)

Concertos

• Beethoven – Violin Concerto (Oistrakh)
• Chopin – Piano Concertos Nos. 1 & 2 (Zimerman)
• R. Schumann – Piano Concerto (Grimaud)
• Brahms – Piano Concerto No. 1 (Freire) (Zimerman)
• Brahms – Piano Concerto No. 2 (Zimerman)
• Brahms – Violin Concerto (Oistrakh)
• Sibelius – Violin Concerto (Oistrakh)
• Scriabin – Piano Concerto (Ugorski)
• Rachmaninoff – Piano Concerto No. 3 (Weissenberg)
• Martinů – Harpsichord Concerto (Růžičková)

Chamber music

• Beethoven – String Quartet No. 13 (Jasper Quartet)
• Beethoven – String Quartet No. 14 (Ariel Quartet)
• Schubert – String Quartet No. 14 (Integra Quartet)
• Mendelssohn – Piano Trio No. 1 (Beaux Arts Trio)
• Mendelssohn – String Quartet No. 6 (Ébène Quartet)
• Chopin – Cello Sonata (Mørk/Stott)
• R. Schumann – Piano Quartet (Goerner/Kashimoto/Karni/Gabetta)
• R. Schumann – Piano Quintet (Vassilenko/Belcea Quartet)
• C. Schumann – Piano Trio (Atlantis Trio)
• Franck – Piano Trio No. 1 (Trio Novanta)
• Brahms – Piano Trio No. 1 (Pires/Dumay/Wang)
• Brahms – Piano Trio No. 3 (Katschen/Suk/Starker)
• Brahms – Piano Quartet No. 1 (Fauré Quartet)
• Brahms – Violin Sonata No. 3 (Oistrakh/Richter) (Roussev/Uinskas)
• Brahms – Cello Sonata No. 1 (Mørk/Lagerspetz)
• Brahms – Clarinet Quintet (Leister/Brandis Quartet)
• Brahms – Clarinet Trio (Trio Légende)
• Tchaikovsky – Piano Trio (Kissin/Bell/Maisky)
• Dvořák – Piano Trio No. 4, 'Dumky' (Beaux Arts Trio)
• Dvořák – Piano Quintet No. 2 (Haefliger/Takács Quartet)
• Elgar – Enigma Variations (Kaspszyk)
• Rachmaninoff – Cello Sonata (Lugansky/Capuçon)

Solo music

• Beethoven – Piano Sonata No. 17, 'Tempest' (Korstick)
• Beethoven – Piano Sonata No. 29, 'Hammerklavier' (Pollini)
• Chopin – Piano Sonata No. 2 (Pogorelich) (Zimerman)
• Chopin – Piano Sonata No. 3 (Zimerman)
• R. Schumann – Piano Sonata No. 1 (Perahia)
• R. Schumann – Piano Sonata No. 2 (Argerich)
• R. Schumann – Piano Sonata No. 3 (Glemser)
• Brahms – Ballades Op. 10, Rhapsodies Op. 79 (Gould)
• Bortkiewicz – Lamentations and Consolations (Somero)
• Medtner – Piano Sonata No. 2, 'Night Wind' (Eckardstein)

Short pieces and movements

• Beethoven – Coriolan Overture (Karajan)
• Beethoven – Grosse Fuge (Dover Quartet)
• Schubert – Impromptu Op. 90, No. 3 (Horowitz)
• Schubert – Piano Sonata No. 21, Mov. I (Zimerman)
• Mendelssohn – String Quartet No. 2, Mov. I (Dover Quartet)
• Chopin – Prelude Op. 25, No. 15 (Horowitz)
• Chopin – Nocturne Op. 48, No. 1 (Weissenberg)
• R. Schumann – Piano Trio No. 3, Mov. I (Munich Dvořák Trio)
• Wagner – Lohengrin Prelude (Halasz)
• Brahms – Piano Trio No. 2, Mov. II (Pires/Dumay/Wang)
• Brahms – Cello Sonata No. 2, Mov. III (Mørk/Lagerspetz)
• Brahms – String Quartet No. 2, Mov. I (Dover Quartet)
• Sarasate – Zigeunerweisen (Gitlis)
• Fauré – Cantique de Jean Racine (Letonja)
• Fauré – Pavane (Handford)
• Fauré – Sicilienne (Isserlis/Devoyon)