Learn Russian: Мы должны учить лексику и правила.

This is the main rule from this lesson! We use учить when talking about memorizing specific things, like vocabulary (лексику) and rules (правила). You would use изучать for studying a broad academic subject (like history), and учиться for where or how you study (like at a university).

Why is должны used for 'have to' instead of a normal verb?

Russian does not have a regular verb for 'must' or 'have to'. Instead, it uses the short adjective должен. Because the subject is the plural Мы (we), it takes the plural form должны. It links directly to the infinitive verb учить.

What case are лексику and правила in, and why?

They are in the accusative case because they are the direct objects of the verb учить (the things being learned). The feminine noun лексика changes its ending to -у to become лексику. The neuter plural noun правила looks exactly the same as its dictionary form because inanimate neuter nouns do not change in the accusative.

Is лексика related to the English word 'lecture'?