In The Taming of the Shrew, what is the relevance of the Induction to the plotline?

This is an excellent question. It is rather amusing to
note that so many directors choose to willingly omit the Induction from productions of
this excellent play even when a clear relationship between the Induction and the rest of
the play can be drawn. If you are studying this play, I think it is incredibly important
to realise how Shakespeare uses the convention of the play within a play in this comedy
just as he does elsewhere in his oeuvre (for example in A Midsummer Night's
Dream).

To my mind, the relationship between the
Induction and the rest of the play has two central correlations: the way that it points
towards the artificiality of the play setting, especially concerning cross dressing, and
the way it indicates the importance of social class in Renaissance
England.

Firstly, I am sure you are aware that during
Shakespeare's time, no women were allowed to act. Thus the part of girls were always
played by young males or boys whose voices hadn't broken yet. We need to recall that
sexuality was very different in those days, and such gender "confusion" slotted in well
to a time when homosexuality was not necessarily the taboo or vilified subject that it
is today. However, it appears that Shakespeare is deliberately pointing towards the
artificiality of the play by having the Lord's servant playing the role of Sly's "wife."
Likewise, the banter that goes on between Sly and the characters of the Induction was
typical of the banter that would occur between the players and the "groundlings" or
those that stood up to watch the performance and were literally inches away from the
actors.

Secondly, it is noteworthy that the Induction
contains, in order of importance, a member of the ruling class, the middle class (the
innkeeper) and the working class (Sly). Shakespeare points towards the tension between
the classes with the rather witty dialogue between Sly and the Innkeeper and of course
the whole idea of convincing Sly that he is a Lord indicates the way that the Lord is
staging his own play where the "swine" Sly as he is referred to is allowed to assume the
role that he says is his by right of ancestry. Of course, the theme of disguise, assumed
identity and mistaken identity is rife in all of Shakespeare's comedies, and here this
motif is indicated once more.