This is actually quite simple. I've never been particularly good with this rule, so I recently looked up when to use each.

The answer is that whenever you would use "he" or "she", you would use "who". If you would use "him" or "her", use "whom".

I'll give you some examples.

"To whom are you writing this letter?" "I am writing to her."

"Whom will you be arriving at the dance with?" "I will be arriving with him."

"Who is that?" "She is the new girl."

Pretty easy, right?