I have learned that "kiu" means "who" if you are asking a question. What word do you use if you are not asking a question? Suppose I want to say, "I am the person who is studying Esperanto." Is "kiu" appropriate there? What about "whom?" Suppose I want to say, "This is the man whom taught my Esperanto." What word is a good translation for "whom?"

Foreigner:I have learned that "kiu" means "who" if you are asking a question. What word do you use if you are not asking a question? Suppose I want to say, "I am the person who is studying Esperanto." Is "kiu" appropriate there? What about "whom?" Suppose I want to say, "This is the man whom taught my Esperanto." What word is a good translation for "whom?"

In thy (your) first question, "kiu" is the word to use. This is called the "relative pronoun" usage. In the second sentence it should still be "kiu" (and even "who" in english since it's the subject of the verb "taught") but in the sentence "this is th man whom I taught Esperanto" "al kiu" would be used; what's actually being said is "this is the man to whom I taught esperanto.'

Strictly speaking, you should never say "This is the person whom taught my Esperanto." You should say "This is the person who taught me Esperanto."

Exactly as the others have said. Note further, that the word that is generally equivalent to "whom" is "kiun".

The woman whom I kissed = La virino, kiun mi kisis.

However, English and Esperanto differ in a few key places for the usage of "who/whom". For example, "whom" (like "me", "him", etc.) is used after prepositions in English, whereas Esperanto says that the preposition tells us enough about the usage of a word without the accusative-n:

The man to whom I gave my hat = La viro, al kiu mi donis mian ĉapelon.

(Just like "I gave it to him" = "Mi donis ĝin al li", notice there is no "n" on "li", even though the English uses "him" instead of "he" ).