Nice set of recommendations. Newbee, Gary17 and Beemerrider: thanks for posting such extended and excellent suggestions for specific works as starting points for people beginning to explore classical music. I agree with the suggestions.
My contribution here will be to reinforce the suggestion that one start with borrowing some music from your local library. Sample some of these works at no cost and see what you connect with. And, if you have a classical music radio station (or webcast) available to you, spend some time with that as background music while you do other things.
Also, remember that each of us may connect via different styles of classical music. For a friend of mine, the normal starting points I'd suggested (e.g., Mozart) just didn't connect for him. But, when he heard some Shostakovich, that resonated immediately for him. Shostakovich is not where I'd normally think of starting someone on a classical journey, but this music really spoke to him and nailed him as a classical music listener. Now he's branching into other things.
So, for those who are just not getting into classical from the "normal" points of entry, here are some 20th Century music recommendations to consider in case your tastes are a bit more twisted and you need something a bit more out of center to catch your interest (not normally the best starting point for most people, but for some...):
Shostakovich, Sym 5 (Previn/LSO, RCA) and String Quartet No. 8 (Borodin Qt or Fitzwilliam QT, Decca)
Kodaly, Hary Janos Suite (Kertesz/LSO, Decca)
Varese, Ionisation (Mehta/LAPO/LAPercussionEnsemble on Decca or NJ Percussion Ensemble on Nonesuch)
Bartok, Music for Strings, Percussion & Celeste (Reiner/CSO, RCA or Solti/LSO, Decca)
Crumb, Vox Balaenae (Voice of the Whale) or Ancient Voices of Children
Larsson, Concertino for Bassoon & String Orch, op45,4
Lutoslawski, Cello Concerto (Lutoslawsk/OrchParis, Rostropovich -cello, EMI)
Panufnik, Concertino for timpani, percussion & strings or Sinfonia Rustica
Poulenc, Concerto for Organ, Strings & Timpani (Pretre/FNRO, Durufle -org, EMI)
Schoenberg, Five Pieces for Orchestra, op16 (Dorati/LSO, Mercury)
Schuller, Seven Studies on Themes of Paul Klee (Dorati/MinneapolisSO, Mercury)
.