Learn Spanish (Latin America): La niña bebe jugo y come yogur.

Because the subject is la niña (she), which is the third-person singular. For regular -er verbs, you drop the -er from the dictionary form and add -e. If you were talking about yourself, they would end in -o (bebo, como).

Why isn't it el jugo and el yogur?

Just like in English, when you talk about an unspecified amount of a food or drink, you don't need to use "the" (the definite article). You simply say bebe jugo (drinks juice) and come yogur (eats yogurt).

I've seen the word zumo used for juice. Is jugo the same thing?

Yes! Jugo is the standard word for juice in Latin America. Zumo is primarily used in Spain. Since you are learning Latin American Spanish, jugo is the word you will hear and use the most.

Since yogur doesn't end in -o or -a, how do I know its gender?

Yogur is a masculine noun, so you would say el yogur. It is a borrowed word from another language, and in Spanish, these types of loanwords usually default to being masculine.