Learn Russian: Мама варит суп на кухне.

Готовить is the general word for preparing food or cooking. However, Russian uses варить specifically for boiling or making things in liquid, like soup, pasta, or coffee. So, you usually use варит when talking about making soup!

Is суп in the nominative case here?

It is actually in the accusative case because it is the direct object (the thing being cooked). Because суп is a masculine, inanimate (non-living) noun, its accusative form looks exactly the same as its dictionary (nominative) form.

Why does it say на кухне instead of в кухне for "in the kitchen"?

In Russian, the word кухня (kitchen) is a common exception that almost always takes the preposition на instead of в to mean "in" or "inside". You will also see this exception with words like почта (post office) and работа (work).

Why does кухня change to кухне at the end of the sentence?

It changes to кухне because it is in the prepositional case, which is used to show location (answering the question "where?"). For feminine nouns ending in -я, the ending changes to -е in the prepositional case.