There is no way I could, in a single effort, list the most brilliant or important musical works of all time so this is but one installment of suggested listening and/or supplemental music history.
Wolfgang Amadeus Mozart - String Quartet No 19 in C, K465 'Dissonant Quartet' - A glimpse into the future from 1785
Mozart - Symphony No 40 in g minor K550 (1788)
Mozart - Symphony No 41 in C 'Jupiter' K551 (1788)
Ludwig van Beethoven - Piano Sonata No.14 in c# minor, Op. 27 no. 2 'Moonlight Sonata' (1801) - Everyone should know the first movement already but the third movement is the truly amazing one in my opinion.
Beethoven - Piano Sonata No.23 in f minor, op. 57 'Appasionata' (1805)
Richard Wagner - Tristan und Isolde (1859) - While it's probably an exaggeration to call Tristan the first work to defy tonality, it was the first work to do so for over 4 hours. Wagner refuses to resolve his dominant 7th chord and reappropriates the tension state into the release state laying the groundwork for both referential structuralism (neo-tonality) and dodecaphony. Furthermore, this piece was supposedly written from the heart, not the head as the inversionary set relations with a 50% invariance factor would appear to suggest.
Antonin Dvorak - Symphony no. 9 in e minor, 'From the New World' (1893) - Dvorak's American symphony, hence the name since he was a Czech, stands as one of the greatest romantic symphonies alongside those of Schubert and Tchaikovsky.
Claude Debussy - Prélude à l'après-midi d'un faune (Prelude to the Afternoon of a Faun), L.86 (1894) - Perhaps the most well known and, arguably, greatest piece in the expressionist vein. Debussy displays his complete mastery of subtle orchestration.
Alexander Scriabin - Piano Sonata No.5 in F#, Op. 53 (1907) - Scriabin's first piano sonata in the single movement form.
Anton Webern - Fünf Sätze für Streichquartett, Op.5 orchestrated (1909)
Arnold Schoenberg - Fünf Orchesterstücke in der Originalfassung for orchestra (Five Orchestral Pieces), op. 16 (1909 rev. 1922) - Schoenberg's post-tonal pre-dodecaphonic expressionist masterpiece
Ralph Vaughan Williams - Fantasia on a Theme of Thomas Tallis (1910) - You've probably heard this or a cue derived from this composition in several films.
Igor Stravinsky - The Rite of Spring (1912) - Nearly a century later, it still stands as the greatest polytonal work ever written. This is, arguably, the most important piece of music from the 20th Century.
Scriabin - Piano Sonata No.9 'Messa Noire (Black Mass)', Op. 68 (1913) - If No.5 is not Scriabin's greatest work, then this is.
Maurice Ravel - Orchestration of Mussorgsky's Pictures at an Exhibition (1922)
Alban Berg - Wozzeck Op.7 (1922) - Berg's adaptation of George Buchner's Woyzeck is, to this day, the greatest atonal opera ever written.
Webern - Symphonie, Op. 21 (1928) - Webern takes formalized structuralism to a new level with his palindromic and total serial approach.
Dmitri Shostakovich - Lady Macbeth of Mtsensk, op. 29 (1932) - In what is perhaps the greatest opera of the 20th century, Shostakovich takes musical characterization to a whole new level.
Vaughan Williams - Symphony No.4 in f (1934) - Vaughan Williams BACH piece that raises the bar for cel expansion and motivic development. This is easily Vaughan Williams most dissonant and powerful work. He hated it.
Berg - Violin Concerto "Dem Andenken eines Engels (To the Memory of an Angel)" (1935) - Dedicated to a Boston native Louis Krasner, Berg's final finished work is generally accepted to be his greatest. Berg blurs the lines seperating tonality from dodecaphony to create a romantic masterpiece. Berg immortalizes himself...a giant who lived in a box.
Shostakovich - Violin Concerto No.1, Op. 77 (1948) - Not heard until over a decade after it's composition, Shostakovich's First Violin Concerto is truly a symphony by all indications and, in my opinion, his greatest.
Iannis Xenakis - Metastaseis (1954) - Xenakis' first orchestral work is the beginning of stochastic composition. It is tied, mathematically, to the Philips Pavillion of Le Corbusier designed for the 1958 World's Fair in Brussels. Varese's Poeme electronique was composed for performance in that building.
Krystof Penderecki - Threnody for the Victims of Hiroshima (1959) - Just like Xenakis' Metastaseis, Threnody further expands the sonic paradigm for string orchestra.
Shostakovich - String Quartet No.8, Op. 110 (1960) -
Just read this. Xenakis - Synaphaï (1969) - This is, essentially, Xenakis' piano concerto. The piano part is notated across 10 staves with performance directions instructing the pianist to play as many of the lines as possible.
Alfred Schnittke - Concerto Grosso No.1 for two violins, prepared piano, harpsichord and 21 strings in six movements, Op. 119 (1977) - Schnittke defines, for me, the late Russian realist paradigm with this piece.
Schnittke - Concerto for Piano and Strings (1979) - This work is a soul-shattering vengeful and mournful masterpiece of unparalleled expressionism and craftsmanship.
Ok, that should keep you all busy for awhile. I will attempt to fill in the (many) gaps in a month or two.