Learn Russian: Послезавтра они едут в город. - The day after tomorrow they are driving to the city.

Russian often uses the present tense to talk about firmly planned future events, just like English! Saying они едут (they are driving) with a future time word like послезавтра makes it clear that the plan is already set.

Could I use идут instead of едут here?

You would only use идут if they were walking to the city on foot. Because cities are usually far away, едут (going by vehicle or transport) makes much more sense here.

Why doesn't the word город change its ending if it is the destination?

When you show motion toward a place, you use в + the Accusative case. For inanimate masculine words ending in a consonant, like город (city), the Accusative case looks exactly the same as the basic dictionary form. So it doesn't need a new ending!