Being new to classical as I am at the moment is a very confusing place to be. The repertoire is huge and there are 7 million performances of every piece. After much random listening Bach and Janacek seem to offer things that appeal to me as a starting point.
Janáček is an excellent place to start. I guess you know the two string quartets? Even in the early 70s, these were regarded as odd, peripheral to the main repertoire, and too demanding to play because of their unusual Czech sounds. Now theyre pretty much standard fare for most aspiring string quartets. Some of the earlier recordings on Supraphon (eg by the Janáček quartet) are very good indeed. The Janáček quartets are also a good starting point for many of the well-known string quartets of the 20th century (Bartók, Shostakovich, Berg, Dvořák (late 19C, but he is Czech)), and also the less well-known (eg Smetana, Haas, Weinberg, Ahmed Adnan Saygun). If you dont know Weinberg, but like Janáček, Bartók or Shostakovich, try one of the Danel Quartets recordings on cpo you might be surprised. You could then work backwards to the great 19th century quartets (Beethoven, Schubert), or even forwards to the second half of the 20th century (look at the repertoire of the Arditti Quartet as a guide). A little digging on the internet will usually throw up which recordings are good value for money sites like La Folia, or Paris Transatlantic, are pretty good sources of commentary on contemporary music. Or you could listen to the radio Radio 3 is fine, but I presume that European radio stations are generally available from UK satellite boxes? If so, France Musique, SWR2, WDR3 or Bayern-Klassik are at least as good as Radio 3, and often much better.
You could do something similar with Janáčeks operas, in following the line backwards and forwards, listening to interesting pieces as they appear on the radio.