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.