Learn Spanish (Latin America): El mesero nuevo sirve la comida.

Sirve comes from the verb servir (to serve). It is a "stem-changing" verb. This means that when you conjugate it in the present tense, the e in the middle of the word changes to an i for most forms. Since "the new waiter" is a single person (he), we use the "he/she" form, giving us sirve.

If it's a stem-changing verb, would the "we" form also change to an "i"?

No, the "we" (nosotros) form is an exception. Stem changes almost never happen in the "we" form. If we were saying "we serve the food", it would be nosotros servimos la comida, keeping the original e from servir.

Why does nuevo come after mesero?

In Spanish, descriptive words (adjectives) like nuevo (new) usually come after the thing they are describing (the noun). So, instead of saying "new waiter" like in English, you say "waiter new" (mesero nuevo).

I have seen the word camarero used for waiter. Is mesero different?

They mean exactly the same thing! Mesero is the most common word for "waiter" in Latin America, while camarero is more commonly used in Spain.