Well...
The Pogues are actually pretty simple to get into. They recorded a total of five albums with their original lineup (the first three being the really "essential" ones) and those five are constantly being reshuffled into various "best ofs", which usually cover the most important stuff.
Their main strength is in the songwriting of Shane MacGowan, which is shown to best advantage on their second album "Rum Sodomy and the Lash" (also their best-sounding album) and third "If I Should Fall From Grace With God" (probably the best collection of MacGowan originals, and one of the albums that never leaves my mp3 player). Both albums are lifeblood to me (I grew up listening to the Clancy Brothers).
Their first album "Red Roses For Me" would be the next purchase. It's nearly as good, but I don't think the originals are quite as outstanding as on the next two.
There was an EP released between albums two and three, called "Poguetry in Motion", which features what is arguably MacGowan's best song ("The Body of an American", featured prominently in the series finale of The Wire), as well as two more terrific originals and a nice band instrumental. These turn up on various CD collections (though the original 4-song CD is probably long out of print).
Those three albums plus EP have been the core of the band's live set in the two decades plus since their release (they seem to reunite every few years, so long as MacGowan can stand, though he really looks like hell these days).
Albums four and five ("Peace and Love" and "Hell's Ditch") show the band moving slightly away from their trad roots, but they suffer from the Creedence effect: a genuinely genius songwriter is forced (by health or just a need to appease the troops) to record originals by the rest of the band, and with rare exception, there's nothing to touch his best stuff. So I recommend both albums, but only after you have the others.
The Pogues continued on without MacGowan for a couple of pretty forgettable albums (I've even forgotten the names), and they're not unlistenable, but as I say, not memorable either.
There's quite a bit of one-off and B-side material that turns up on CD reissues and/or collections. Their Motown one-off "Yeah Yeah Yeah Yeah Yeah" is a real hoot, and was nearly a hit in the U.S. I'm also partial to some of their traditional non-album tracks (notably "The Limerick Rake" and "The Parting Glass") which again sometimes turn up on various reissues or anthologies.
As I say, MacGowan's drinking is wearing him out pretty fast (though I suppose you could also argue that the fact that he's even still alive is a testament to its preservative powers), but with his subsequent band, The Popes, he has put out two kickass studio albums ("The Snake" and "The Crock of Gold") and a great live set called "The Rare Oul' Stuff". There's also a decent live album of a Pogues reunion show from 2001 called "Live at Brixton Academy." Avoid the one called "Streams of Whiskey"-- not only was it released without the band's permission, but the sound isn't nearly as good as the Brixton release.
I was thinking of going to see them again a couple years back, when they were playing San Francisco's Fillmore... but the tickets were $75. The last time I saw them at The Filllmore, tickets were $15! (yeah, yeah, I know. I'm old!)
ETA: To get really technical, only the first two Pogues albums feature the original lineup. After album two, bassist Cait O'Riordan went off to have kids with Elvis Costello. She did one lovely lead vocal on album two ("I'm A Man You Don't Meet Every Day"), but apart from that, the change wasn't really significant.
