Identify the kinds of knowledge Hamlet values; discuss the ways in which he acquires makes use of that knowledge.

I'll have a go at your question.  Hamlet, without a doubt,
is intelligent, perceptive, and insightful.  This fact is evident throughout the play.
 As such, he seems to value and use different kinds of
knowledge.

First of all, he is an empiricist.  He is the
"observe'd of all observers."  Through his careful observation of others, he is able
deduce Claudius' excessive drinking, his mother's too sudden fall for Claudius,
Ophelia's betrayal, Polonius' and other of king's spies' espionage, and Claudius' guilt
(as he reacts to the Players). He makes use of these observations as he navigates
throughout the corrupted currents of the court, deftly managing Claudius' attempts to
contain him.

Hamlet also possesses cultural knowledge:  he
knows the Bible well (Notice his allusions in his conversation with Polonius in Act 2);
he knows Greek and Roman mythology; he is well versed in drama; he is astute in history
and current events.  Hamlet uses this knowledge to set a trap for Claudius, to elude
Polonius, and to connect with Horatio.

He is also rational
and logical.  Look at his most famous speech, "To be or not to be."  It is a well
reasoned argument, posing and answering the question of why we "bear the whips and
scorns of time" when we could our "quietus make with a bare bodkin."  The speech is a
beautiful piece of rhetoric, which outlines the pains of living, the temptation of
suicide, and the fear that causes us to cling to that which we know rather than to
travel to that "undiscovered country, from which no traveller returns."  Other examples
of his logical mind can be seen when he shows Claudius in Act 4, Scene 3 how a "king can
progress through the guts of a beggar," and in Act 4, Scene 4, when he analyzes the
nature of thought, man, and action and the relationship of all three.  More examples of
this type of analytical thinking occur in Act 5 in the graveyard scene and much earlier
in Act 2 with Rosencrantz and Guildenstern, with Hamlet's speech "What a piece of work
is man."  Perhaps it is this knowledge that Hamlet most values.  And perhaps it is this
type of thinking that leads to certain philosophical conclusions about action, man,
death, life that leads us to judge Hamlet as a prince of thought rather than a prince of
action.

Most of all, perhaps, Hamlet is intuitive.  He lets
his conscience guide him, which at times causes his most pronounced conflicts.  He is
torn between his love and loyalty to his father and his obvious reluctance to murder.
 He attempts to operate with conflicting values, attempts that have tragic
consequences.