Oh, you certainly can go looking for depth in his work – and you’ll even find it…Just not in the Elric books, or in his heroic fantasies in general (at one point, he wrote so fast that he could write a whole book in a weekend – and it shows). If someone wants serious Moorcock, try The Brothel in Rosenstrasse, Gloriana (my favorite) or Mother London.
The series of omnibuses, as published in the US (the UK titles, order and contents are somewhat different) are:
The Eternal Champion – the UK edition is better, since it collects the three “John Daker” books together, but Daker/Erekose is one of the better EC avatars, and the odd novel thrown in here (The Sundered Worlds) isn’t bad, though it doesn’t really fit.
Von Bek – contains two not-exactly heroic fantasies from the '80s about Ulrich Von Bek, which are quite good and somewhat more weighty than pure hack-and-slash, and the third John Daker book (which was in volume 1 in the UK, where it belongs).
Hawkmoon – four novels set in a far-future fantasy Europe; this is probably Moorcock’s least interesting series. It’s pure adventure, but pretty unexciting; I’d leave this for last, or not read it at all.
A Nomad of the Timestreams – three alternate-worlds novels about Oswald Bastable, featuring various odd and implausible alternate histories that are closely related to late 19th century British adventure fiction. (Fans of the comic The League of Extraordinary Gentlemen would probably like this.)
Elric: Song of the Black Sword – collects the first three (by internal chronology) Elric novels as of the time the omnibus was put together. I think the Elric novels should really be read in the order of publication (as others have said, Moorcock killed off Elric early, so practically all of the other novels are flashbacks in the first place), but that can be hard to do, since they’ve been reshuffled so many times.
The Roads Between the Worlds – Three weird SF novels that have very little, if anything, to do with anything else in the series. Worth reading if you like Moorcock’s '60s writing style, or want pulpy SF adventure.
Corum: The Coming of Chaos – the first trilogy about Corum, last don’t-call-him-an-elf, who goes off to kill three gods (the same ones Elric worships, interestingly). I personally like Corum better than Elric, because he’s otherwise very similar but whines less, but this series is less popular than the albino’s.
Sailing to Utopia – Four more novels that really don’t belong; at some point Moorcock obviously saw this series as a way to get his whole backlist into print. What I said about Roads also applies here.
Kane of Old Mars – a very pulpy trilogy that is also a very obvious homage to Edgar Rice Burroughs. Fun if you like that sort of thing, but not much to do with the EC.
The Dancers at the End of Time – one of Moorcock’s better series, though also very unrelated to the EC. It’s about decadent immortals at the end of time, and features some of his best writing. (It’s light-hearted and fun, too.)
Elric: The Stealer of Souls – the rest of the Elric novels as of the early '90s. It has Stormbringer in it, the best of those novels and also the book in which Elric kicks the bucket.
Corum: The Prince With the Silver Hand – another trilogy about Corum, in which he wanders off to another world and has to save it, too. Fun but a bit fomulaic, though still miles above Hawkmoon.
Legends from the End of Time – collects two books related to Dancers – one weird novel and a collection of good short stories. Well worth checking out if you liked Dancers.
Earl Aubec – an unabashed short story collection, without much to do with the EC.
Count Brass – the second Hawkmoon series, which is better than the first.
The UK series also had, somewhere in the middle, The New Nature of the Catastrophe, an anthology of stories edited by Moorcock, all about Moorcock’s anti-hero Jerry Cornelius (who has his own series, mostly in the literary/confusing mode, which intersects the EC mega-series in occasional and tangental ways).
Basically, Moorcock has been trying to tie every single word of fiction that he’s ever written together for the past two decades or so. A few books still roam free, but most of his work is, really or ostensibly, tied together in some way. So, if you find that you like his writing, it can be fun to dive into it and see how random things connect with each other. But if you only like one side of Moorcock (either the unabashed speed-writer of fantasy adventure or the would-be literary lion), it can be tougher to pick out the stuff you’ll like.