Learn Russian: Нет, я не учитель, я студент.

Нет means "no" and is used at the beginning to answer a yes/no question. Не means "not" and is placed right before the specific word you want to negate. Here, you use нет to give your overall answer, and не to show that you are not a teacher (не учитель).

Why is there no word for "am" in this sentence?

In Russian, the present tense of the verb "to be" (am, is, are) is almost never used. Instead of saying "I am a student," you simply say "I student" (я студент). The same rule applies to "I am not a teacher" (я не учитель).

Would a female speaker say this exact same sentence?

Not exactly. Учитель (teacher) and студент (student) in this sentence are masculine forms. A female speaker would change the nouns to match her gender, saying учительница for "teacher" and студентка for "student".

How should the intonation sound when speaking this sentence?

Because this is a statement giving an answer, your voice should drop in pitch at the end. You also need a slight pause after Нет, just like after "No" in English, and another pause at the second comma before stating what you actually are.