Hi Chris,
First things first: if you're purely interested in expanding your musical horizons, so to speak, then there's no particular reason to confine yourself to EMI Great Recordings of the Century (hereafter GROC), and no particular reason why if you liked that you'd like any others in the series - the recordings on GROC are simply deemed to be significant in some artistic / historical sense and are very diverse in terms of repertoire. To be fair there are a lot of very good records in that series, but equally there are a lot of very good records outside it, and if you're just on the lookout for nice cello music to listen to then basically any classical label will have lots of great stuff to choose from.
With all that said, that combination of cellist du Pré and conductor Barbirolli does have a certain legendary status, in particular for
this recording of the Elgar cello concerto, also on GROC, which really is an outstandingly excellent record and certainly well worth owning. I'm a big fan of the
Brahms cello sonatas too, though I haven't heard that recording and the music isn't quite such a guaranteed hit as - for example - the Elgar cello concerto, which is liked by basically everybody in the world.
Of the ones I've heard, my two personal favourite GROC releases are also two of my top ten all-time favourite CDs, namely the big-hearted Rostropovich / Oistrakh
Brahms Double Concerto and Beethoven Triple Concerto (both of these works feature parts for solo cello BTW), and
Elizabeth Schwarzkopf singing Strauss's Four Last Songs, which for me is pretty much the most devastatingly beautiful recording ever made of anything.
WRT the music on that CD you've been listening to. Time-wise you're in the Classical period, ie. around Mozart, after Bach but before Beethoven, and a very long time before Elgar. Genre-wise you're dealing with the cello concerto, ie. the combination of solo cello with orchestra. Depending on how you listen, you might find that other cello concertos float your boat, in which case the Elgar above would be a good choice. The other super-popular concerto in the cello repertoire is the Dvorak, which IMHO is the finest of them all. For a beginner I'd heartily recommend just diving in and exploring the repertoire available;
this set covers a fair amount of ground in 5 CDs at £2 per disc, including core works like the Dvorak and Shostakovich concerti, Tchaikovsky's
Rococo Variations and alternative versions of the Haydn works, and is by all accounts very good throughout (
review and full track listing here). [I suspect you don't want at this stage to be getting into the business of buying multiple versions of the same music, so in that sense it's a shame that the above set includes recordings of the Haydn you've already got, but if you're a big fan then you might find it interesting to compare alternative interpretations.]
However, the likes of the Dvorak and Elgar - the two most popular cello concertos - are thoroughly Romantic as opposed to Classical, written around a century after the stuff you've been listening to and hence completely different in many ways; more immediately heart-on-sleeve expressive, perhaps less clearly structured and in some respects more difficult to get to grips with. You might find that, rather than exploring cello concertos through the ages, you'd be better off listening to lots of other Classical-period music - in which case, grab yourself something like a bunch of the late Mozart symphonies, the piano concertos or the Requiem and dive in.
The ultimate destination of the serious cello purist's pilgrimage through musical history is
Bach's suites for unaccompanied cello. Personally I wouldn't recommend that for the classical beginner - the intensity and austerity of the sound and the rigorous concentration of the music make it rather hard going IMO and with no lush orchestral sonorities to bask in there's no sonic respite.
Hope some of that is useful! - do let us know how you get on.
. I did make a range of "if you'll like that, you'll like this..." type suggestions too, and you'll notice I did also make your suggestion of exploring by period rather than by genre.
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