Learn Russian: После перерыва мы продолжаем этот процесс. - After the break, we continue this process.

It changes because of the preposition после (after). This preposition always requires the noun following it to be in the Genitive case. The dictionary form is перерыв (a masculine noun ending in a consonant), so we add -а to make the Genitive form: после перерыва.

How would I say "before the break" instead?

You would use the preposition до (before/until). Just like после, до requires the Genitive case. So you simply swap the preposition and keep the exact same noun ending: до перерыва (before the break).

Why don't этот процесс get special Accusative endings, since they are the direct object?

They actually are in the Accusative case! For masculine, inanimate nouns (non-living things like процесс) and the words that describe them (like этот), the Accusative form is identical to the Nominative (dictionary) form. They only get special Accusative endings if they are animate (living beings).

Is продолжаем an imperfective or perfective verb?

It is the imperfective verb продолжать (to continue), conjugated here for мы (we). We know it must be imperfective because the sentence describes an action happening right now in the present tense ("we continue" / "we are continuing"). Perfective verbs in Russian only exist in the past and future tenses.