First thing for you to know is that who/whom is a pair of pronouns just like he/him and she/her. There's a quick and easy trick that I'll tell you about later, but you get to actually learn first.
We'll start with 'who.' Use 'who' when you are referring to the subject of a verb. (The subject of a verb is the person or thing that is performing the action. Never use 'who' when referring to a thing.)
Who paid for the meal?
I didn't see who stole your newspaper.
I wonder who is upstairs. (the verb is "to be"; i.e. "who is")
Now for 'whom.' Use 'whom' when you are referring to the object of a verb. (The object of a verb is the person or thing that is receiving the action. Never use 'whom' when referring to a thing.)
Whom should I vote for?
To whom did you give the letter?
You know exactly whom I'm talking about.
Note: Always use 'whom' after prepositions such as 'to', 'with', 'by', 'on', 'in', 'near' etc.
Note: These rules also apply to the use of 'whoever' and 'whomever'.
And I said above, who/whom is simply a pair of pronouns. So just swap them out!. 'Who' is used where a nominative pronoun could also be used; if you can substitute 'I', 'we', 'he', 'she', 'you', or 'they' for it, then 'who' is correct. 'Whom' is used where an objective pronoun could be; if you can substitute 'me', 'us', 'him', 'her', or 'them', then 'whom' is correct.
Whom did you see leave? - Did you see him leave?
Who did you say left? - Did you say he left?
Who gave us the day off? - He gave us the day off.
Seems easy enough, usually. But when you get into longer and more complicated sentences it requires more forethought.
He knows who/whom to call. - He knows to call him. [use whom]
The members who/whom have paid their dues are qualified to vote. - Members are qualified to vote if they paid their dues. [use who]
Who/whom did you say called? - Did you say he called? [use who]
She was a person who/whom the politicians could not influence. - The politicians could not influence a person like her. [use whom].
So just stop and think about it; many intelligent people pause while they're talking. They're thinking before they speak!
Knowing when to use 'whom' is more than most people can manage. And if people rarely use 'whom' in spoken English, they quickly become disinclined to use it in written English. Clumsy speech influences us more than careful writing does.
If 'whom' sounds "pretentious" in spoken English, it's only because we seldom hear it used. And though 'whom' is on the endangered list, it endures here and there, among those who still value elegance in language.
Show 'whom' the respect it deserves. Use it correctly. :)
Grammar Monster: Who & Whom
The American Heritage Book of English Usage
Dr. Grammar: Who or Whom?
The Dictionary of Disagreeable English