Learn Russian: Заяц бегает быстро.

Бегать is a multidirectional verb of motion. It means the hare runs around in general, back and forth, or that running is a habit. If we wanted to say the hare is running in one specific direction right now, we would use the unidirectional verb бежит.

How would we say "I see the hares" to practice this lesson's focus on plural animate objects?

Заяц is an animate noun. For animate nouns, the plural accusative (used for direct objects) looks exactly like the genitive plural. The fleeting vowel я changes to й, and we add -ев because the stem ends in -ц and the ending is unstressed. This gives us зайцев. So, "I see the hares" is Я вижу зайцев.

Could I use быстрый here instead of быстро?

No, because быстро is an adverb describing the verb (how the hare runs). Быстрый is an adjective, which you would use to describe the noun itself, like in the phrase быстрый заяц (a fast hare).

Why does the я in заяц change to й when we add endings to it?

The я in заяц is a "fleeting vowel." When we add endings to change the grammatical case, this vowel drops out of the word's stem. Because it originally followed another vowel (а), it turns into the consonant й to bridge the sounds and make pronunciation smoother.