Mozart,Haydn,Beethoven String Quartets......

You can't go wrong, although tastes will vary, with recordings by the Tokyo, Emerson, or Alban Berg quartets of this repertoire in the digital era; some like the Lindsays a lot (particularly British listeners) in Haydn. For period performances of Mozart and Haydn, you could try the excellent Quatour Mosaiques. If you can get over the need for digital recordings, the best performances bar none IMO are the Vegh for the Beethoven, and the Tatrai for Haydn. I also like the Quartetto Italiano in Mozart. For Haydn, don't bother with the generally critically lauded Kodaly performances on Naxos, an admittedly attractive budget alternative-- they just aren't that good, and aren't a patch on the older Tatrai.

I have the Tokyo Quartet for Beethoven and Alban Berg for the others. The only drawback to the Emerson Quartet is they record for Deutsche Grammophon. In general, I don't care for how DG's recordings sound.

For the Beethoven, Cleveland Quartet. ( By the way they do one of the best Debussy/Ravel) Can't make any high recomendation for the Mozart, maybe the Budapest S. Q. on #'s 14-19. There is the Heutling Quartet on EMI import from France. Not sure how they sound, but you get the complete quartets + 4 quintets. Tweekerman

I heartedly second Swklein's recommendation of recordings by the Mosaiques Quartet. Their Haydn is excellent, as are the recordings by the Kodály Quartet on Naxos.

My recommendations:
HAYDN
String Quartets Op. 76, Nos. 1-6
Kodály Quartet (Naxos)
Mosaïques Quartet (Naive Astrée)

BEETHOVEN
Tokyo or Emerson or Alban Berg Qts

MOZART
String Quartets, K387, K421(Haydn Quartets, disc 1)
Mosaiques Quartet (Naive Astrée)

String Quartets, K464, K465 ("Dissonance") (Haydn Quartets, disc 2)
Mosaiques Quartet (Naive Astrée)

I'm a little surprised that these recordings have not been recommended yet, but for the Beethoven String Quartets, I think the Guarneri Quartet's recordings are in a class by themselves.

There's also an interesting recording of the Mozart Quartets by the American String Quartet. The recordings were done using the "Axlerod" Stradivari instruments that are housed at the Smithsonian. The playing, and the sound, is fantastic.

Just my two cents. Regardless, I hope you enjoy listening to all of these great, great works.