Learn Russian: Я быстро выучу это правило.

This is the core of the simple future tense! The verb выучить is a perfective verb, meaning it focuses on a completed action. When you conjugate a perfective verb using the normal present-tense endings (like the -у for я), it actually creates the future tense.

Why is the ending -у instead of -ю for the я form of выучить?

Выучить is a second conjugation verb, so it drops the -ить to add its endings. Normally, the я form would take -ю, but Russian has a spelling rule: after the letter ч (and certain others like ш, щ, ж), you must write у, never ю.

What case is это правило in this sentence?

It is in the accusative case because it is the direct object (the thing being learned). However, because правило is a neuter noun and inanimate (not alive), its accusative form looks exactly the same as its basic dictionary form. The same goes for the pointing word это.

How is выучить different from the basic verb учить?

Учить focuses on the process of studying or learning, while выучить (the perfective form used here) focuses on the result. By using выучу, the speaker is saying they will completely memorize the rule, not just spend time looking at it.