Considering how little we know of Shakespeare as an individual, it is impossible to definitively determine what his purpose was in writing The Taming of the Shrew. Seeing as it is a comedy, one of Shakespeare's earliest comedies at that, one can assume Shakespeare primarily wanted to entertain his audience with the antagonistic antics of Katharina and Petrucchio, then give them a satisfying finale by having the two wed.
The play ends with Katharina submitting to her husband's will and becoming the ideal wife, reinforcing Elizabethan ideas about gender and marriage—or so it would seem. In recent times, some have speculated that Katharina's final submission is not what it seems to be. For example, film versions such as the 1929 Sam Taylor film and the 1967 Franco Zeffirelli movie suggest that Katharina's submission is more a facade and that she only pretends to submit, while still remaining very much her own woman. (The 1929 movie even has her literally winking at the audience in the middle of the speech!)
However, whether or not this is the modern world's way of salvaging the play from its problematic content or an intended irony from the Bard himself is hard to say. In any case, the play does showcase an interest in gender roles and the relationship between men and women. It is up to the individual audience member and/or reader to decide if Shakespeare is reinforcing male domination or subverting it.