If introspective, gloomy, angry, dramatic verging on cacophonous, the audio equivalent of every human tragedy, disaster and stroke of violent idiocy - not to mention the feelings of alienation and resentment of Authority found in so much of heavier popular music - are your thing, I wouldn't recomment that much of Mahler. Or Bruckner, or any of the other heavier, more florid romantic Germanic compsers.
Rather, try something with a little more attitude, a little more craziness and a little less melodrama. Shostakovich's later symphonies, particularly numbers 8, 10 and 11. Prokofiev's more inane works. In my experience the Russians do this stuff a lot better than the Germans, and often more noisily too. Admittedly they are often a challenge to listen to, they completely defy the "Oh, isn't that nice and relaxing, like a brain massage, it makes me sleep" line spouted by so many people who've never listened to any classical with balls.
I was raised on classical music from birth and love a lot of it, but I admit that much of it does waffle on in the pretentious manner that those who don't know its glories associate with its stuffiness and frilly old-people conservatism. To escape that, if you don't like the violence of the 20th century Russians, what I cannot reccommend enough are Bach's solo works for piano or violin.
These pieces have a purity, an austere elegance verging on almost mathematical minimalism. More than anything, the most perfect piece of classical music I have ever heard is the Ciaccona from Bach's Partita No. 2 for solo violin. A few other people have also stated this view, including Yehudi Menuhin, who declared in one optimistic conversation that it was capable of curing most evil in the human mind. The best recording I have heard is by Itzhak Perlman, but there are no doubt many other excellent versions out there, I come from a family of cellists and our collection of violin recordings is sadly lacking.
By and large, though, I am disheartened by the number of people who honestly try to get into classical, but only listen to the fluffy, light, well known to the point of cliched stuff. Particularly that which comes on those discs that are my greatest hate: "The Best of Beethoven!" "Popular Classical!"< rant > NO! YOU FOOLS! Don't break a symphony or a sonata into bits! It just DOESN'T WORK! It's not finished! You can't put half a song on a CD, don't put half a concerto! I can't listen to those CDs, they lurch from snippet to snippet with no conclusion of any of the pieces, hopping illogically from key to key and style to style until your brain gives up and runs out of your ears if you try and listen. No wonder it puts people to sleep. No wonder they look at me pityingly when I say I like classical. All I can say is: buy the whole damn piece. And buy at least one thing a little more hardcore than "The Swan" from Carnival of the Animals or "The Less Interesting Movements of the Four Seasons." < /rant >
A list of what I have found non-classically inclined friends have considered cool:
As for John Williams, I've found in the music for Star Wars: Episodes 1 and 2 a quite astonishing degree of similarity to so much of Prokofiev's better stuff. Ah, geniuses think alike, eh? Actually he gets more Russian as he gets older, he was verging on Shosty on some of the score for Minority Report.