I see Inge made some good suggestions on the music front. Ignoring Beethoven and Mozart, here are a few more. Many of these have been used in films.

Elgar: Enigma variations, plus the Pomp and Circumstance marches. March number 1 famously had lyrics added ("Land of Hope and Glory" etc) but what is little known is that Elgar profoundly disapproved of this, especially given the horrors of World War I at the time, and came to despise the nationalistic fervour it inspired.

Bach: Suite No. 3 in D Major and the Saint Matthew Passion.

Samuel Barber's Adagio for Strings.

Handel: Sarabande from the Keyboard suite in D minor.

Schubert: Piano Trio in E- Flat (second movement)

Vivaldi: Cello concerto e-minor (third movement)

Strauss: Blue Danube Waltz. Try to listen to it without picturing spacecraft from 2001: A Space Odyssey. Speaking of which, if you want something a bit more avant-garde, try Ligeti's Atmospheres. :)

Finally, if you really want something properly melancholy, check out Funeral Canticle by John Tavener.

I enjoy the odd bit of Wagner, but to paraphrase Woody Allen, after a while I do get the urge to invade Poland. Also, I see I've left out Chopin, which is silly of me. So many I could recommend, but I'll leave it there for now. :)