Learn Russian: Я покупаю торт.

It actually is in the Accusative case! However, because торт is a masculine inanimate noun (a non-living thing ending in a consonant), its Accusative form looks exactly the same as its dictionary form.

How would I say "I am buying cakes" using the plural rules from this lesson?

You would say Я покупаю торты. To make the regular masculine noun торт plural, you add -ы to the end. Because "cakes" are inanimate objects, this Nominative plural form is also used as the direct object.

Does покупаю mean "buy" or "am buying"?

It means both! Russian doesn't have a separate "am doing" tense. Я покупаю translates to either "I buy" or "I am buying", and you just use the context to tell which one makes the most sense.

There is no word for "a" in the Russian sentence. How do we know it's "a cake" and not "the cake"?

Russian doesn't have articles like "a", "an", or "the". Whether торт means "a cake" or "the cake" is completely understood from the context of your conversation.