Start with WQXR, the classical music station of the New York Times. You can listen to it online. Use their online schedule to find out what you like.
Get some of the bare essentials:
Bach: Brandenburg concertos, Goldberg variations, Violin concertos
Beethoven: Piano sonatas Appasionata, Moonlight and Pathetique and symphonies 3/5/7/9
Chopin: Nocturnes and preludes
Dvorak: Cello concerto, Symphony 9, Slavonic dances
Haydn: String quartets Op. 76, symphony 94
Mozart: Requiem
Rachmaninov: Piano concertos
Schubert: Trout quintet
Stravinsky: Rite of Spring
Verdi: Get a CD of arias
Vivaldi: Four seasons
Pick up a book like
The NPR Guide to Building a Classical CD Collection
or
The New York Times Essential Library: Classical Music.
If you want to buy music use
ArkivMusic.
Bump this thread when you find something you like and some us can suggest similar pieces.