My personal favorite composers (in no particular order) are Mahler, Bruckner, Beethoven, Bach, Copland, Berg, Bartók, Stravinsky, Debussy, Chopin, Brahms, Schubert, Shostakovich, Mozart, Hindemith, Glass, Schoenberg, Prokofiev, Haydn.
If I'm not listening to orchestral works, then I'm usually listening to string quartets or solo piano (I don't care much for harpsichord (very limited dynamics for my taste) or "historically accurate" pianofortes (these sound like out-of-tune19th century whorehouse pianos to me)).
Occasionally I'll listen to operas by different composers, or lieder by Schubert.
My favorite recordings:
George Szell - Beethoven Symphonies cycle
Jenny Lin - Shostakovich: 24 Preludes And Fugues, Op. 87 (Vladimir Ashkenazy does a pretty good one, too)
John Wilson, BBC Philharmonic - Copland: Orchestral Works, Volume 1 - Ballets
Stanislaw Skrowaczewski, Saarbrücken Radio Symphony Orchestra - Bruckner: Symphony No. 9 In D Minor, WAB 109
Carlo Maria Giulini, Chicago Symphony Orchestra - Mahler: Symphony No. 9 In D
Glenn Gould - Bach: Goldberg Variations, BWV 988 (1954 recording)
Boston Symphony Chamber Players - Stravinsky: The Soldier's Tale (with John Gielgud, Tom Courtenay, and Ron Moody); Octet For Wind Instruments; Ragtime
Claudio Abbado, Vienna Philharmonic - Mahler: Symphony No. 4 In G (with Frederica Von Stade)
Claudio Abbado, Chicago Symphony Orchestra - Prokofiev: Lieutenant Kijé - Symphonic Suite, Op. 60
Diogenes Quartet - Schubert: Complete String Quartets
Neeme Järvi, Scottish National Orchestra - Prokofiev: The Complete Symphonies
Leonard Slatkin, St. Louis Symphony Orchestra - Mussorgsky: Pictures At An Exhibition, Night On Bald Mountain, Khovanshcina (MFSL)
Aram Khachaturian, Vienna Philharmonic Orchestra - Khachaturian: Spartacus, Gayaneh; Glazunov: The Seasons
Herbert Blomstedt, San Francisco Symphony Orchestra - Hindemith: Orchestral Works
Jeroen van Veen - Glass: Complete Piano Etudes
Etc. etc. etc.
If you'd like a little more info about musical works and their composers, you can't go wrong with The Vintage Guide to Classical Music by Jan Swafford. A very easy read to get you grounded in classical music.
(If you prefer more detailed information, try getting the latest edition of Grout's History of Western Music as a used book (it's a fairly technical college textbook though). Another great reference for classical music of all kinds is the Harvard Dictionary Of Music.)
If you stick with it, I think you're in for one of the greatest adventures of your life. Have fun and good luck.