Learn Russian: Иностранец изучает русский язык.

Because язык (language) is a masculine inanimate noun. As you are learning in this lesson, masculine nouns only take a new ending in the Accusative case if they are animate (people or animals). Since a language is not alive, язык looks exactly like its dictionary form here.

Does the adjective русский also follow this 'inanimate' rule?

Exactly! Adjectives have to match the noun they describe. Because язык is an inanimate masculine noun in the Accusative case, the adjective русский also keeps its dictionary form. If the foreigner were studying an animate Russian person, both the noun and the adjective would change their endings.

Why use изучает instead of учит for studying?

Both verbs can mean to study or to learn. However, изучать (here as изучает) is usually used when you are studying an academic subject, a science, or a language in depth, like taking a course. Учить is often used for memorizing things or more casual learning.

Is there a trick to remembering the word иностранец?

Yes! Look for the word страна (country) hidden right in the middle of it. The word is built from an old root for other plus страна plus -ец (a suffix for a person, similar to -er in English). It literally translates to a person from another country.