|Well, the different styles generally correspond to the different periods in time: Baroque, Classical, Romantic, and the plethora of styles that is Modern. For most people, I think, the best place to start would be in the Romantic period, because that's where all the big heart-string pullers are, such as Schubert (check out his Unfinished Symphony as well as the Death and the Maiden Quartet), Beethoven (even though he's technically Classical, not Romantic; any of the odd-numbered symphonies should do it), Tchaikovsky (Symphonies 4-6, Piano Concerto #1, and Violin Concerto #1), Brahms (Symphony No. 1), etc.|
If you want something lighter, try Mozart (the later Piano Concertos + last three Symphonies) or Haydn (string quartets Opus 76), from the Classical period; if you want something heavier (and LOOOOOONGER), try Bruckner (9th symphony) or Mahler (6th or 9th, or the 1st, with its perverse Frere Jacques), both of whom straddle the edge between Romantic and Modern.
In the Modern era, there are all the hyper-dramatic and romantic Russian composers, such as Shostakovich (Symphony No. 5, 10), Rachmaninov (Piano Concertos Nos. 2 and 3), and Russia's prankster, Prokofiev (Violin Concerto No. 1, Piano Concerto No. 3). Then there're the impressionists, most important of which is Debussy (La Mer, String Quartet); and the expressionists, such as Stravinsky (Le Sacre du Printemps), Bartok (String Quartets Nos. 4,5: music to murder by), Schoenberg, and Berg (who honestly wrote a pretty beautiful Violin Concerto). After Berg, I don't listen to much, nor do I listen to much Baroque...maybe others could fill you in on the minimalists and serialists.
My personal favorite classical CD of all time is a recording of Jacqueline du Pre playing both the Dvorak Cello Concerto and, more importantly, the Elgar Cello Concerto. It's pretty heart-wrenchingly fantastic.