Learn Russian: Завтра директор расскажет новости. - Tomorrow the director will tell the news.

You've spotted the main trick of the simple future tense! The verb рассказать (to tell) is a perfective verb, meaning it focuses on the completion of an action. Because an action can't be completed in the ongoing present, adding present-tense endings to a perfective verb automatically makes it future tense.

What is the infinitive of расскажет, and where did the ж come from?

The infinitive is рассказать. When you conjugate this verb, the consonant з changes to ж before the endings (я расскажу, он расскажет, etc.). This kind of consonant swap is very common in Russian verbs to make them easier to pronounce.

In English, "news" is usually singular (like "the news is good"). Is новости singular or plural here?

It is plural! This is a big difference between English and Russian. In Russian, новость (feminine) is a standard countable noun that means "a piece of news." In this sentence, the director is sharing multiple pieces of news, so the plural form новости is used.

What case is новости in since it is the thing being told?

It is in the accusative case, because it is the direct object receiving the action of the verb. Since "news" is an inanimate object, its plural accusative form looks exactly the same as its plural nominative (dictionary) form.