At this stage in the proceedings, I should perhaps point out that there is a quite superb Fall discography online. It's right here, at the unnoficial Fall website and if you go to the singles section and then to Hit The North, the scan of the cassette single cover is my scan!!!!!!!!!
The cassette single has version 4, which is awesome.
The Complete Peel Sessions box set is now the universally accepted necessity, both for old fans and new. You'll get 24 sessions (nearly 100 songs!) from 1978 all the way to 2004. It is, simply, the beast career overview of any band, ever. And it's good value too - in England it's about £25 (@ $35?). The versions here are also diffferent from the LP versions (and often superior in my opinion - Athlete Cured being a great example).
It's difficult to recommend LP's, but the box set will make you happy and give you a feel for the band's different eras - with 25 years, and lots of members, there's variety in there.
If I were
to recommend LP's, then I'd have to say Hex Enduction Hour, if only because it is so popular amongst fans. Not one of my favourites though.
Room To Live, Bend Sinister, The Infotainment Scan, Cerebral Caustic, The Unutterable, Fall HEads Roll.
Good live LP's include Seminal Live and Live at Phoenix, although 27 Points is being re-released sometime soon, so maybe wait for that (it's not too great though).
There's so much that they've done - it's worth going to the excellent Unnoficial Fall Website and browsing there.
There's studio LP's, live LP's, compilations, compilations of alternative versions, about 90 LP-style releases in all! Browse the LP part of the dicsography.
And get the Peel Box Set!