Which parts of the talk between Polonius and Reynaldo in Act II, Scene i and I.iii of Shakespeare's Hamlet are "ironic" and why?I've been having...

This is a very interesting question topic to analyze. 
Polonius is usually regarded as a conniving man whose primary concern is with outward
appearances and reputation.  The irony of these two conversations comes from the fact
that what we might expect from Polonius is the opposite
of what we get in the scene with Reynaldo
later.

To review:  When Polonius gives his laundry list of
advice to Laertes in Act 1 scene 3, the common theme of the advice is look out for his
reputation.  He tells Laertes to not dress in too gaudy a manner, to keep his opinions
to himself, to not get into fights -- but if in one -- to win; to be friendly -- but not
too friendly.  He ends the list with the admonishment "this above all to thine own self
be true."  While it may mean that he wants Laertes to just "be himself" I think he
intends it to mean to remember to always put himself first and to maintain his "self"
meaning his outward appearance.

The irony of the scene with
Reynaldo is that he specifically suggests to Reynaldo that the best way to get
information about Laertes is to spread small lies about him and then judge how the
people react to such information. He says that "by indirections, find directions out." 
For example, he should say "Wow, that Leartes is quite a gambler."  Then he should wait
and see what people say in response.  If they say, "No way! He is never playing games."
Then they will know that Laertes to staying away from gambling; but if they say, "You
are so right -- he loses a fortune every night!" then they know that Laertes is, in
fact, gambling.  While this plan may work, the problem is that even if the people deny
the negative comments, the comments have still been made and people might start to
believe them.  These comments will potentially tarnish what might be Laertes pristine
reputation!  Even Reynaldo questions the wisdom of this plan, but Polonius assumes that
Reynaldo will "season it in the charge" meaning he will say it in just the right way so
as to not create a scandal, but that is still a very real
possibility.

For a man who is so clearly concerned with
reputation, he is ironically taking quite a risk with this plan.