These are my recommended works for J.S. Bach:
ORCHESTRAL
- Brandenburg Concertos [6 of them], BWV 1046-1051
- Orchestral Suites [4 of them], BWV 1066-1069 (Suite No. 2 contains the famous Air on a G String)
- Double Violin Concerto in D Minor, BWV 1043
CHAMBER
- 6 Suites for Unaccompanied Cello, BWV 1007-1012
- 6 Sonatas and Partitas for Unaccompanied Violin, BWV 1001-1006 (Partita No. 2 contains the famous Chaconne)
KEYBOARD (on piano or harpsichord):
- The Well-Tempered Clavier (Book I and II), BWV 846-893
- Goldberg Variations, BWV 988 (get the recording by Glenn Gould, he is the best)
ORGAN
- Toccata and Fugue in D Minor, BWV 565
CHORAL
- Mass in B Minor, BWV 232
- St Matthew's Passion, BWV 244
- Magnificat, BWV 243
These are all good starting points to exploring the works of Bach but I can say that virtually anything Bach composed is worth hearing.
I am not a really big fan of Mozart so I don't know much about his works. But from my (rather limited) Mozart collection I can recommend:
- Clarinet Concerto in A Major
- Piano Concerto No. 21 in C Major, "Elvira Madigan"
- Any of his late symphonies (>35 maybe?). I like the Symphony No. 41 in C Major, "Jupiter".
Bach and Mozart composes during the Baroque and Classical era respectively, so you may wish to listen to other composers who live in the same era.
Bach's contemporaries:
(Frankly speaking their music doesn't really sound like him).
- Handel (His most famous work is the oratorio "Messiah". Also try the oratorio "Saul in Egypt". For instrumental works, try the Water Music and Royal Fireworks Music).
- Vivaldi (His "Four Seasons" is wildly popular. His works are very consistent in quality, so you can just sample any of his other concertos).
Mozart's contemporaries:
- Haydn
The Classical age ends around the nineteenth century, but a number of composers still continue to write in the Classical style.
Classicists after Mozart/Haydn:
- Brahms (Symphonies, concertos)
- Mendelssohn (A Midsummer Night's Dream, especially the overture, Hebrides overture, Symphony No. 4, Songs Without Words)
- Saint-Saens (Piano Concerto No. 2, No. 4, cello concertos, violin concertos, Carnival of the Animal, Symphony No. 3)
You can search for recordings at amazon.com or http://www.allmusic.com. Better still, get yourself a buying guide like Penguin, Gramophone, or Third Ear. (I only own the Penguin to be honest).
If you are just starting to listen to classical music may I also recommend these books:
- The Rough Guide to Classical Music
- Classical Music: 50 Greatest Composers and 1,000 of Their Greatest Works by Phil G. Goulding
- The Essential Canon of Classical Music by David Dubal (this last one will still be useful even for old-time collectors).