To put an end to the question of which composer to go with (if, for some bizarre reason, you could only pick one) let me just say:
Bach=God
Could the God Emperor be wrong? Nope.
As for specific pieces, lets start with these.
The Organ Toccatas and Passacaglia
Performed by Christopher Herrick on the Hyperion label. I read a review that said, "If you only own one disc of organ music, this must be it." Having several organ discs already, with recordings of all these pieces, I was a bit skeptical. And I was wrong. What an
amazing disc!!
Brandenburg Concertos
Hard to go wrong with performances. Hogwood, Harnoncourt, Tafelmusik are all good performances on period instruments. (Meaning they're playing 300+ yo instruments, or replicas.) But Bach sounds like paradise if you play it on kazoos, so just go for it.
Cantata #140 - Wachet auf (Sleepers, wake)
Conducted by Karl Richter. Soloists: Edith Mathis, Peter Schreier, Dietrich Fischer-Dieskau. Recorded 1979, I think. Label: Deutsche Grammophon
Cantata #4 - Christ lag in Todes Banden (Christ lay in the bonds of death)
Condunted by Richter. Soloists: Kieth Engen, Hedwig Bilgram. Recorded 1959 or 63. Label: Teldec
Goldberg Variations
Performed by Hantai on the Opus 111 label, or Pinnock on Archiv.
St. Matthew Passion
Conducted by Herreweghe or Gardiner. (More expensive multi-disc, but might be mid-priced by now.)
St. John Passion
Gardiner
Sonatas and Partitas for unaccompanied violin
Performed by any of the following:
Perlman
Szeryng
Mintz
Unaccompanied Cello Suites
Performed by any of the following:
Ma
Rostropoich
Wispelwey
Kirkpatrick
Let me know when you need more.
And if you start threads asking the same about Beethoven, Schubert, or Bartok, I'll give you some suggestions.

I'm not particularly fond of Mozart, but I know a bit there too.