You wouldn't normally listen to just one part ("movement") and rarely to more than a couple of pieces at a time. Classical pieces are usually numbered ("opus", abbreviated "Op.", which is Latin for "work"). Schubert is one of those composers who didn't number them himself. He died young of syphilis and left his manuscripts in a disorganised mess. They were sorted through by a guy called Deutsch, hence the numbers here are "Deutsch" numbers. Here, "Death and the Maiden" is D. 810.
The Italian after the Roman numerals for the movements are speed markings ("allegro" = "fast"), and this piece shows a typical progression: fast, slow, playful/fast, faster. They should be listened to in order, one after the other. The idea is that the music takes you through a sequence of moods, like courses in a dinner.
Try listening to string quartets. Here are a few brilliant ones:
All of Schubert: https://www.youtube.com/playlist?list=OLAK5uy_nK-_o1OYLszr8d...
Shostakovich 8: https://youtu.be/HgExfvXq7VI
Beethoven 10 (Harp): https://www.youtube.com/watch?v=exMaWKVcCEs
Mendelssohn 2: https://www.youtube.com/watch?v=6UAU7xbTOyA
There are so many more but string quartets are such a wonderful place to get started. Intimate, pure music, with so much passion and excitement.