How do Benedick and Beatrice insult each other, mainly in act 1, scene 1, but throughout the play as well?

As Leonato in the first scene of this play, in reference to Beatrice's facetious queries as to how many men Benedick has killed and eaten, "there is a kind of merry war betwixt Signior Benedick" and his niece. This scene serves to give the audience an insight into the nature of the relationship between the two, and its structure allows us to see that Beatrice "insults" Benedick habitually when he is not present, as at the very beginning of the scene, and Benedick also speaks critically of Beatrice to his friends, as at the end of the scene. It is not only when they are together that the "skirmish of wit" takes place: both Benedick and Beatrice make sure to continue their war even when they are not together.
Of course, their preoccupation with thinking up witty criticisms of each other leads those around them to question what this interaction really implies, and whether the insults are truly meant as such. Benedick remarks on Beatrice's "fury" to Claudio, but he also says that she "exceeds" Hero "as much in beauty as the first of May doth the last of December." In this scene, Shakespeare is setting up his hero and heroine as two people who very much enjoy pretending to dislike each other, but the foreshadowing at the end of the scene from Don Pedro—"I shall see thee, ere I die, look pale with love"—is an early suggestion to the audience that, before the play is over, the two will be reconciled.

The insults the pair direct to each other, importantly, are intended to amuse, not only each other but those around them, as Leonato has evidently recognized. Beatrice makes a comment that would not be out of place in a romantic comedy to this day when she says, "I wonder that you will still be talking, Signior Benedick: nobody marks you." Benedick responds by calling her "my dear Lady Disdain," and Beatrice's reciprocal quip is that disdain cannot possibly die while it can feed on such fit matter as Benedick provides. The interchange is like a parry, or a dance: Beatrice offers an outright insult—"scratching could not make it worse, an 'twere such a face as yours were"—and Benedick calls her "a rare parrot teacher." The audience is supposed to interpret that this is an argument that has been much played out between the two of them, and it certainly is playing. Note how Beatrice says that Benedick "always end[s] with a jade's trick," alluding to a poorly bred horse (jade) which cannot finish a race. Between Benedick and Beatrice, their witty interchanges are a competition, and Beatrice resents being denied the full extent of them.

We know that the interchanges between these two are born out of a long association. We see this also in act 2, scene 1, in which Beatrice is completely unfooled by Benedick's disguise and happily insults Benedick to his face, but pretending she does not know it is him, calling him "a very dull fool" in order to see his reaction. In seeing through Benedick so quickly, however, what Beatrice is really telling the audience is that the two of them are, in their own way, close; they know each other and they know what to say to rouse each other's humor and anger. Through the wordplay of their insults, Beatrice and Benedick enact their own kind of courtship.