What secret plots or schemes are in Act 2 of Hamlet, and what are their desired outcome or effects?
In Act 2 of a Shakespearean play, things usually become more complicated, and that is certainly the case with Hamlet. This entire play is about secrets and plots, so all these start to manifest themselves in Act 2. The act opens with Polonius instructing an agent named Reynaldo, on how exactly to spy on his son Laertes, who is away at school. Polonius is very concerned about the family reputation, and wants to ensure that all is well, but his methods leave something to be desired in that he wants Reynaldo to tell small lies about Laertes in the hopes that Laertes's friends will react in a telling way as to whether the "lies" are true or not. That is one scheme.
A second more important scheme is Hamlet's act of madness which he told us about at the end of Act 1. We first learn of this scheme when Ophelia reports to her father about Hamlet's strange behavior, but we see it for ourselves shortly after when Hamlet appears at court and seems to not know people and talks in strange riddles that even Polonius comments on saying, "though this be madness, yet there is method in it." Hamlet's scheme is to have everyone think he is crazy so that perhaps Claudius will let down his guard and reveal his guilt in the murder of King Hamlet.
Related to this madness is a small scheme by Polonius and Claudius to test whether Hamlet is truly mad because of the lost love of Ophelia. Polonius is convinced he is, but Claudius is skeptical and wants some proof, so the two men decided to set-up a meeting between Ophelia and Hamlet and watch what happens from behind some curtains. They are plotting to discover the truth of Hamlet's madness.
A final scheme that is planned at the end of Act 2 is Hamlet's plot to have the traveling acting troupe perform a play for the court which he "will have changed some dozen of sixteen lines" in so as to more closely match the actual events of King Hamlet's murder. Hamlet hopes that viewing this play will twinge Claudius's guilt and that "the play's the thing wherein I'll catch the conscience of the king."