Normally I'm against this because I feel that a music piece should be heard in its entirety; not just single movements or fragments. Luckily, there are many single movement pieces out there I can recommend to you:
Chopin:
Ballade No. 1
Ballade No. 4
Barcarolle
Polonaise Op. 40 No. 2
Any Nocturne
Liszt:
Consolation No. 3
Ballade Ukraine
Ballade No. 1
Vallee d'Obermann
Scriabin:
Fantasie Op. 28
Sonata No. 5
Sonata No. 7
Alkan:
Le Vent from Op. 15
RV Williams:
Fantasia on Thomas Tallis
Rachmaninov:
Isle of the Dead
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Now here are some movements taken from larger works:
Mahler:
Symphony No. 5 Mov. 4
Symphony No. 6 Mov. 3
Symphony No. 1 Mov. 1
Bach:
Aria from the Goldberg Variations
Contrapunctus XIV from The Art of Fugue
Chaconne from Violin Partita No. 2
Rachmaninov:
Piano Concerto No. 2 Mov. 2
Sonata No. 2 Mov. 2
Ravel:
Ondine from Gaspard de la Nuit
Prokofiev:
Romeo and Juliet, the Balconey Scene
Symphony No. 5 Mov. 1
Sibelius:
Pelleas et Mellisande: At the Castle Gate
Beethoven:
Piano Sonata No. 32 Mov. 2
Piano Sonata No. 8 Mov. 2
Mozart:
Concerto for Flute and Harp Mov. 2
That's enough to get you started.