How does one know when to use "who" and "whom"?

This is a good question; and most people try to get it
right but often don't because they somehow think whom is more
sophisticated and use it more often than it should be
used. 

I'm assuming you're at least familiar with the
basics of grammar and understand what an object is in a grammar discussion. 
Whom is used as a pronoun replacement for any object:  direct
object, indirect object (though not a likely usage, it's correct), or object of the
preposition.  That's a lot of choices, I know; so, it may be easier to remember that
who is used to replace subjects and predicate nominatives.  So, for
example, in the following sentences, we use
who: 

I voted for the candidate
who is stronger. (subject of the adjective
clause)

Who is going to the movie?
(subject of the sentence)

Who?  Susie is
who.  (predicate nominative)

The
following sentences use whom as
objects:

Ask not for whom the bell tolls.  (object of the
preposition)

You asked whom to the dance?  (direct
object)

In general terms, if the word who
or whom comes toward the beginning of a sentence, you
would usually use who.  If the word who or
whom comes later in the sentence, it's generally
whom.  That's not always true, and that rule doesn't really work
with questions (or other inverted sentences), but it's a
guideline.

Thank you for at least wanting to use these
correctly.  Hope this helped.