Learn Russian: Какое сегодня число?

Что means "what" when asking about a thing or an action. Какой (and its neuter form какое) is used when asking "which" or "what kind of". Since число (number/date) is a neuter noun, we use the neuter form какое to ask "which date".

Does число literally mean "date"?

Число literally means "number". When talking about the calendar, Russians ask "which number" it is to find out the specific date.

I don't see a verb in this sentence. Where is the word for "is"?

In the present tense, Russian normally leaves out the verb "to be" (am, is, are). So this sentence literally translates as "Which today number?".

Since this lesson is about the future tense, how would I ask "What will the date be tomorrow?"

Because the verb is skipped in the present tense, you have to add the future tense form of "to be" to ask about tomorrow. You would say Какое завтра будет число? using the future tense verb будет (it will be).

Why is сегодня pronounced with a "v" sound instead of a hard "g"?

The letter г in the grammatical endings -его and -ого is always pronounced like an English "v". The word сегодня comes from two older words meaning "of this day", so it uses that special "v" pronunciation rule.