I came to Borges backwards. The Raw Shark Texts led me to House of Leaves which led me to Borges. One story in particular, Blue Tigers, with its mathematically-impossible discovery, reminded me strongly of House of Leaves.
However, this volume, which contains all of Borges' short stories, ranges further and wider than either of those two novels. The stories are grouped according to the eight books in which they were originally published (I was tempted to count this as books 34-41), beginning with A Universal History of Iniquity (1935), and ending with Shakespeare's Memory (1983). Incidentally, the volume most often seen in British bookshops, Labyrinths, isn't here because it's drawn from two separate Borges books, Fictions (1941) and The Aleph (1949).
The ordering makes clear the progression Borges made, from the mischievous blurring of fact and fiction in A Universal History... to testing the limits of the short story in the 1940s and the questioning of the boundaries of reality itself in his later work.
Andrew Hurley's well-researched endnotes add to the stories enormously, providing detail that would otherwise escape a non-Argentinian reader. His translation is very readable, though I've seen it criticised for making Borges' language too modern.
If you like labyrinths and knife fights, you'll love this book. Even if you don't - and I'm a big fan of labyrinths but less keen on the knife fighting - this collection is packed with extraordinary stories.
If you're looking for the best labyrinth story, I'd recommend Ibn-Hakam al-Bokhari, Murdered in his Labyrinth (1949) but you could try The House of Asterion, also from 1949, which is a slightly unusual and strangely affecting story.
Man on Pink Corner (1935) is the first of Borges' stories of slum-dwelling tough guys. It establishes a pattern he returns to in later stories: the narration is one half of a conversation, the events in the story always happened some time ago and the action centres on a critical knife-fight.
In The Garden of the Forking Paths (1941) and Death and the Compass (1944), Borges plays with the detective story, pushing the genre's typical framework in unusual directions. In The Disk (1975) and Blue Tigers (1983) he examines characters drawn to magical, impossible objects. His stories are frequently fantastic, in a literal sense, without being fantasy.
I could list a dozen other dazzling examples from the Complete Fictions, though I wasn't that keen on the two volumes of prose poetry, The Maker (1960) and In Praise of Darkness (1969).
The stories frequently turn in on themselves, like Borges' beloved labyrinths, and the author often upends expectations within collections, recasting a theme in the space of two or three stories. Occasionally Borges even pulls the same trick across anthologies: the hero of Juan Murana (1970) changes the interpretation of Man on Pink Corner.
There are several instances of Borges conjuring a moving twist at the end of a story but more often he's playing with an idea. Borges' stories are intellectual rather than emotional. That's not a problem though, particularly when the intellect on display is as playful and inventive as his.