We learn several things about Sherlock Holmes in this novel. First, this is where we are introduced to "the woman" or Irene Adler. Sherlock is madly in love with Irene, who is a criminal.
We also learn that Adler is, somehow, Sherlock's match. She manages to outwit him just as often as he outwits her. They are to be locked eternally in a match of wits.
We also learn that Sherlock has issues with cocaine, as he is only switching between cocaine and new cases while Watson sets up house with Mary Watson (his wife).
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.