We learn in this story that Holmes is, after all, fallible. In his previous appearances, in the two novels A Study in Scarlet and The Sign of Four he cleared up all points of the case presented to him in a spectacularly successful manner. Here, however, his ‘best plans (are) beaten by a woman’s wit,’ as Watson notes rather wryly. Irene Adler outsmarts him, fools him by a disguise, just as he usually fools others, and leaves the country, taking with her the incriminating photograph which her former lover the King of Bohemia had wanted Holmes to secure. True, she promises not to use the photograph against the King as she is now happily married elsewhere, but it is scant consolation for Holmes.
We also learn in this story that Holmes is certainly not averse to sidestepping the law in order to achieve his object. In this case, he breaks into Adler’s house in his attempt to retrieve the photograph. Watson, too, is more than ready to help him in this.
Perhaps the most striking thing that emerges about Holmes in this story is that he is susceptible to female charms. Watson, always at pains to stress his formidable reasoning powers, emphatically denies that he has fallen in love:
It was not that he felt any emotion akin to love for Irene Adler …. He never spoke of the softer passions, save with a gibe and a sneer.
Even if not actually in the grip of the ‘softer passions’, though, Holmes makes no secret of his overwhelming admiration for Irene Adler. She is intelligent, beautiful, and certainly seems to leave a lasting impression upon the great detective when he asks to keep her photograph – not the photograph that the King had wanted, but another one, just of herself, which she rather cheekily sends with her farewell letter. Never again in the whole Holmes canon does Holmes appear so in awe of a woman.
There are many things that we learn about Sherlock Holmes, one right after the other, in Doyle's "A Scandal in Bohemia," which appeared in Strand and was the first of the Sherlock Holmes stories to be published. The things we learn about Holmes come through the musings and memories of Watson; Holmes's conversations with Watson and others; and Holmes's explanations of his logical observations and deductions. The basis of all that is learned throughout is built in the first few paragraphs of Doyle’s story.
In the first paragraph we learn about Holmes's emotional capacity and his peculiar strain of temperament that makes him the logician and observer that he is:
All emotions ... were abhorrent to his cold, precise but admirably balanced mind. ... the most perfect reasoning and observing machine the world has seen.
We then learn about his "fierce energy" and his unfortunate habit of using cocaine when boredom overwhelmes him--it is interesting in and of itself that Doyle gave this unfavorable trait to Holmes in the strict and rigid Victorian era; one wonders whether Doyle is, as a doctor and writer, pointing out a secret strain of escape from Victorian restraint running through London or simply giving Holmes a very human failing to enhance the character.
In addition, we also learn of some of Holmes's background; e.g., "the case of the Trepoff murder." This is an interesting narrative device employed by Doyle, because, of course, "A Scandal in Bohemia" is the first Sherlock Holmes story and therefore the first Sherlock Holmes case--as he didn't exist previously! We then learn, in relation to Watson's own activities, that Holmes is given to pacing the room when involved in a mystery ("I saw his tall, spare figure pass twice in a dark silhouette") and that he is tall and lean, "spare." We also learn the date of his newest case, "March, 1888," and the location of his abode, "Baker Street," where he and Watson had formerly shared rooms before Watson's marriage, which he associates "with my wooing, and with the dark incidents of the Study in Scarlet" (more of the narrative device establishing a flashback to a background preceding Holmes's creation!).
Finally, in the beginning paragraphs of Doyle's finely crafted story, we learn of Holmes's gift for observing--and trusting his observations--and his gift for deductive reasoning. He observes points of Watson's dress and deduces--correctly!--that Watson has gone back into private medical practice; has gained seven and a half pounds in marriage; has recently gotten very wet and muddy; and has a "most clumsy and careless serving girl." We further learn that Holmes and Watson have a relationship of fondness and mutual respect and good will so that their friendship is indispensable to each other:
Stay where you are [Doctor]. I am lost without my Boswell. And this promises to be interesting. It would be a pity to miss it. ... I may want your help, and so may he.
The remainder of the story adds more details in a similar fashion--woven into the action and the thoughts and the conversations of the participants--to produce a full-blown picture of Holmes and Watson, all built upon the revelations in these first few paragraphs.