Learn Russian: Мы пьём чай на моей кухне.

In this sentence, "kitchen" (кухня) is the location where the action happens, so it takes the prepositional case: кухне. The word for "my" has to match the feminine gender and prepositional case of the noun it describes, so моя becomes моей.

Why do we use на (on) instead of в (in) for "in my kitchen"?

In Russian, you usually say you are "on" the kitchen (на кухне) rather than "in" it. This is a common exception you just have to memorize, similar to saying на работе (at work) or на почте (at the post office).

Чай (tea) is the direct object being drunk. Why didn't its ending change?

You are right that чай is in the accusative case because it is the direct object. However, for masculine inanimate nouns (objects and concepts, not people or animals), the accusative case looks exactly the same as the basic dictionary form.

What is the purpose of the soft sign (ь) in the verb пьём?

The verb пить (to drink) is a bit irregular. The soft sign in пьём acts as a separator. It tells you to pronounce the п and the ё distinctly (like "p-yom") rather than blending the consonant directly into the vowel.