Learn Spanish (Spain): Yo sirvo la sopa y tú sirves el pescado. - I serve the soup and you serve the fish.

The verb servir (to serve) is a stem-changing verb. In the present tense, the e in the stem changes to an i for most forms, giving us sirvo (I serve) and sirves (you serve). The e only stays the same for the "we" and "you all" forms, like servimos.

Is it necessary to include the words yo (I) and tú (you) here?

You could leave them out because the verb endings already tell us who is acting, but keeping them adds emphasis. Spanish speakers often use subject pronouns when contrasting two people's actions, just like emphasizing "I am doing this part, and YOU are doing that part" in English.

Why is the word for fish pescado here instead of pez?

In Spanish, pez refers to a live animal swimming in the water. Once a fish is caught and prepared for eating, it becomes pescado. Since you are serving it as a meal in this sentence, pescado is the correct word.