In the play how do I explain "puns and epigrams." And where are they?

Check your prompt.  Are you being asked to define "pun" and "epigram" and to give examples from the play?  Are you being asked to explain the purpose of "pun" and "epigram" in depicting characters or actions?  Are you being asked to explain why this particular play conveys so much information through the use of "pun" and "epigram"?  Once you identify the question or problem you can  better frame your response.

If your prompt asks you to develop a definition and give examples, then you need to examine what a "pun" and what an "epigram" are.  A general dictionary is not the best place to begin such a search and you might want to  focus your search terms, pairing them with literature - Another way to develop a working definition of a term - in this case "pun" - is to figure out what larger type or class of thing it belongs to.  For instance, after you research "pun", try to state what "a pun" is in reference to a larger category - and then try to state in your own word what makes it different from, say, an "epigram."

Once you get a feel for the terms you might see that the title of the play contains a pun and if you read it a certain way, the title can become an epigram as it does when repeated by Jack Worthing at the end of ACT III.

If you are being asked to explain how "pun" and "epigram" define character and action in the play, then you need to be able to identify puns and epigrams.  Epigrams are short statements that convey a lesson or sentiment that one would probably agree was true.  The meaning of a pun, however, might turn on just a single word or phrase, but may not often fulfill the function of the epigram.  In any case, the epigram and pun are both examples of verbal wit - that is, the person who makes the pun or asserts the epigram, is aware of what they are doing.

Why, then, does this play employ so much verbal irony and wit?  Well, we know that this type of irony involves self awareness - to see the world as it really is and then express it, is truly one of the great dramas that mark our physical and intellectual growth. One of the most famous characters of adolescent literature, Holden Caulfield, from "Catcher in the Rye", illustrates this idea.  In the play, "The Importance of Being Earnest" Algernon is probably the most self knowing character - he sees the stuffy world around him and wishes to escape through "bunburying."  His Aunt Augustus, is probably the least self aware person in the play and is happy with things as they are. You might explain how Algernon's many puns and epigrams, are ways in which he tries to explode the conventions of the society that suffocate him.  But Algernon's explosive wit, is counter balanced by Jack's slow and fumbling path to knowledge of the society around him and his development as a person.

Thus the play concludes with an interesting paradox:  if Algernon's wit has exposed all the falseness of the social conventions around him, how does Jack's epigram concerning "the vital importance of being Ernest" reflect the value of maintaining appearances?