In Act 1 of The Taming of the Shrew, which characters assume false identities and why?

The assumption of different identities is central to the plot development and humor in the play. In scene 1, we are introduced to the characters Lucentio, a young gentleman, and his manservant, Tranio. Lucentio's father has sent him to Padua to study. The two men encounter a group of people and listen in on their conversation. The group is made up of five individuals: Minola Baptista (a wealthy resident of the area), his two daughters (the beautiful and charming youngster, Bianca, and her older sister, Katherine), and two gentlemen.  

It becomes apparent that the two gentlemen in this group, Gremio (an old man) and Hortensio (a younger gentleman) are both interested in winning the attention of the attractive and seemingly charming Bianca. Katherine, Baptista's older daughter, comes across as something of a vixen whom her father wishes to marry off before considering any requests for Bianca's hand in marriage. Baptista also makes it apparent that he wants Bianca educated in poetry and music since these are her favorite pastimes.

Lucentio is quite taken by the beautiful young Bianca and expresses his love for her. He and Tranio reason that the best way to win Bianca's heart would be to approach her in the guise of a teacher. The two then decide to switch clothes so that Lucentio may assume his new role, while Tranio takes on his identity. On the arrival of Biondello, another of Lucentio's servants, they inform him that he has to support their charade by addressing Tranio as Lucentio in public and that he has to only refer to their real identities in private.

In scene 2, we are introduced to Petruchio who promises to assist Hortensio in his quest to woo Bianca. Petruchio is prepared to approach Katherine and eventually marry her, giving both Hortensio and Gremio an opportunity to pursue Bianca. Hortensio, just like Lucentio, decides to adopt a disguise to get close to her. He will present himself as a teacher and offer his services to her father.

The frenzy and urgency with which these characters approach the issue of courting Bianca are what makes their actions humorous rather than deceitful. They make such a fuss about the whole thing that their efforts and plans come across as ludicrous. In the final analysis, the reader or audience is amused by their silly shenanigans rather than being disgusted or shocked that they can even consider being so sly and deceptive. 

In Act 1 of The Taming of the Shrew, which characters assume false identities and why?

There are lots of instances of false identities in The Taming of the Shrew, most of them done out of love or lust which makes them fairly comical rather than diabolical. The tricky false identities are those assumed by Katherine and Petruchio--she a shrew and him a crazy tyrant. Those are personas, disguises if you will, and we realize before the play is over that neither is as outrageous as they appear.