FE is right; if you're not bothering about chronology, then any of the major stories (at least) will do. (I make the distinction because some of the minor pieces ("The Frost Giant's Daughter", "The Vale of Lost Women", "The God in the Bowl", etc.) are by no means Howard at his best, and should be read after you've already acquainted yourself with both the character and Howard's (better) writing.
I'm not sure I'd agree, though, that Hour of the Dragon is the best place to start... despite my liking for that novel, I don't think it is Howard's strongest Conan piece, and it does tend to repeat various things he used in the other tales. However, if you do choose to start with that story, I'd go with that edition (or its later counterpart) because it also incorporates the previously-missing epigraph for the tale, a short bit of verse which is quite evocative and sets the mood for the piece very nicely....