Learn Russian: Официант спрашивает туриста.

It is in the Accusative case because the tourist is the direct object (the person receiving the action). For masculine nouns that represent living things (animate), the Accusative case adds an -а to the dictionary form. That is why турист becomes туриста.

Since официант (waiter) is also a living person, why didn't its ending change?

Официант is the subject of the sentence, meaning he is the one doing the asking. The subject always stays in the Nominative case (the dictionary form). The special animate ending only applies to the direct object, which is the person being acted upon.

If I wanted to say "The tourist is asking the waiter," how would I change the sentence?

You would change the endings to show who is doing the action. The tourist becomes the subject in the dictionary form (турист), and the waiter becomes the animate direct object, getting the -а ending (официанта). The new sentence is: Турист спрашивает официанта.

Does спрашивает mean "asks" or "is asking"?

It means both! Russian does not have a separate continuous tense like English. You can translate it as "asks" or "is asking" depending on what makes the most sense in context.