The thing I love about Borges is that he is both a big-idea sci fi writer and a poet. and his poetry definitely shows through in his sci fi. For example, as in "The Zahir". The thought of a person obsessing about the beauty of the shadow of a rose really does it for me.
But I like his big idea stories too. Absolutely "Tlön, Uqbar, Orbis Tertius" may be his masterpiece. But it's dense for an introduction to the author. Contrast this with "The Secret Miracle" which also has a beautiful big idea, but it's so elegantly small. in a sense the whole story fits into just one second.
Finally, I went overboard and calculated the number of books on the shelves in "the library of babel". I remember distinctly that the story made our very real universe feel small for the first time in my life. and to think, I barely scratched the surface. a whole book was published about the math in that one story alone: "The Unimaginable Mathematics of Borges' Library of Babel" by William Goldbloom Bloch.
The Two Kings and the Two Labyrinths is a page long, and you can find it online (html). It is a small example of his style. If you like that one, then you will probably like his other stories. Of his short stories, my personal favorite is The Secret Miracle (it is included in his book Fictions). He also wrote poetry. My personal favorite is The Golem (included in the book The Other, the Self).
The recommendations you´re getting are right on spot:
Fictions (1944)
The Aleph (1949)
The Book of Sand (1975)
I have to say, however, that not all stories in those books are equally palatable for newcomers. If you´d like more specific recommendations on which stories to read first, let me know. I have written a full page with recommendations for beginners on how to start.
You could get his _Selected Fictions_
Then read
The library of Babel
Funes the Memorious
The God´s Script
The Book of Sand
The Garden of Forking Paths
The above are enjoyable for beginners - even though of course there are many more layers of meanings than are first apparent in those stories. But that deeper enjoyment can come later. They´re formidable narratives in their own right.
I'd recommend anyone new to Borges to begin with "The Garden of Forking Paths." It's a sort of spy thriller story but also includes his trademark meditations on time and infinity.
The first short story of his I read was “Tlön, Uqbar, Orbis Tertius”. It remains my favourite of his, and I probably read this story and handful of his others once a year or so.
For other Borges fans, I'll mention the book: The Book of Form and Emptiness by Ruth Ozeki. It references countless of Borges' stories.