Learn French: Est-ce que vous prenez le train pour Paris ?

Prenez is the vous (you) form of the irregular verb prendre (to take). Notice that it drops the "d" from the dictionary form! While the singular forms keep the "d" (like je prends or il prend), the vous form simply becomes prenez.

The English translation says "Are you taking." Where are the words for "are" and "-ing"?

French doesn't have a separate "-ing" tense for actions happening right now. The single present tense verb prenez covers both "you take" and "you are taking." Never try to translate "are" directly in this kind of sentence!

Why does the sentence use pour Paris instead of à Paris?

While à usually means "to" a city, French uses pour (which normally means "for") when talking about transportation leaving for a destination. It helps to think of le train pour Paris as "the train destined for Paris."

Do I have to use Est-ce que at the beginning of this question?

Not necessarily. Est-ce que is a very standard, polite way to turn any statement into a yes/no question. However, in casual spoken French, you could drop it and simply say Vous prenez le train pour Paris ? while raising the pitch of your voice at the end.