Shostakovich was the greatest composer of the 20'th century. He possessed a level of genius equal to that of the greatest composers in history (such as Bach, Mozart, and Beethoven). Shostakovich was able to match these musical giants in the ability to repeatedly come up with absolutely incredible inspirations, writing masterpieces that induce that profound musical emotion of the human brain.
Here are some of my favorite pieces by Shostakovich. These (of course) are a matter of taste. Pieces (or movements) highlighted in yellow are particularly masterful. But all these pieces are warmly recommended. Newcomers to Shostakovich should hear the two Piano concertos, symphonies #5 and #7, piano trio #2, and string quartets #3 and #8.
One crucial piece of advice, from personal experience: The music of Shostakovich (and really, music in general) often resists initial efforts to appreciate it. Usually it is only after 3 or 4 hearings, when some familiarity with the new piece begins to set in, that an experience of musical enlightenment occurs.
Here are a few masterpieces by other composers which should be more widely known: