If you are new to the string quartet genre, I think the best place to start is Haydn: his works are fresh, sprightly, endlessly melodic, and reasonably short. Among the best interpretators are the Quator Mosaiques (try their
Op.76 cycle). If you want to go budget, Kodaly Quartet's recordings under Naxos are a must.
For Beethoven, start with the middle quartets Op.59. His late string quartets have a daunting reputation, but the last one (Op. 132/133) is beautifully meditative.
Among the string quartets by late Romantic composers, the easiest to get your ears to is Dvorak -- he has this magic touch for structure, form and melody. Brahms is of course a grand master, but may not be a beginner's first choice. Somehow Schubert or Mendelssohn don't click for me, but that of course does not mean their works aren't good.
This budget 2-CD will get you started with the better-known quartets by Shostakovich, before you invest in any of the complete cycles.
Terry's
Salome Dance of Peace is extremely long and, at least to me, rambling and not particularly well-structured. Many a time I tried to finish it but I couldn't.
Requiem for Adam might be a better piece to get acquainted to.