Learn Russian: Мы изучаем русскую грамматику.

The verb изучать is used when you are studying a specific subject, like a language or a science, and it requires a direct object (the thing you study). Учиться means 'to be a student' or 'to study at a place' (like at a university) and cannot take a direct object. Since we are stating exactly what we are studying, we must use изучать.

Could we use учим instead of изучаем for this sentence?

Yes, but the nuance changes. Изучать implies a deep, comprehensive, or academic study of a broad system. Учить usually implies memorizing specific information (like vocabulary words or a poem by heart) or doing everyday schoolwork. Because grammar is a complex system that you analyze, изучать sounds very natural here.

Why is it русскую грамматику instead of the dictionary form русская грамматика?

Because we are doing something to the grammar, it is the direct object of the sentence and must be in the accusative case. The dictionary forms (nominative) are русская and грамматика, which are both feminine. In the accusative case, the feminine adjective ending -ая changes to -ую, and the feminine noun ending -а changes to -у.

Does изучаем mean 'we study' or 'we are studying'?