Breakdown of Завтра мы будем начинать следующий проект, а потом будем его заканчивать.
а
and (contrastive)
мы
we
его
him / it (acc)
быть
to be
Questions & Answers about Завтра мы будем начинать следующий проект, а потом будем его заканчивать.
How does the future tense work in this sentence?
This sentence uses the compound future tense, which is made by conjugating the helper verb быть (to be) and adding an imperfective infinitive. Here, we use будем because the subject is мы (we), followed by the infinitives начинать (to begin) and заканчивать (to finish).
Why does the second half of the sentence say будем его заканчивать instead of keeping the two verbs right next to each other?
In Russian, word order is very flexible. While you can absolutely say будем заканчивать его, it is extremely common to place short object pronouns like его (it) right between the helper verb and the main action verb. Both structures are correct and mean the same thing.
Why is the pronoun его used to mean 'it'? Doesn't that mean 'him'?
Yes, его translates to 'him' when talking about a person, but it is also the accusative (direct object) pronoun for masculine things. Because the word for project (проект) is a masculine noun, we must use его to say we will finish 'it'.
Why does the phrase следующий проект (the next project) look exactly like its dictionary form if it is the direct object receiving the action?
Because проект is an inanimate masculine noun (a thing, not a living being). For inanimate masculine nouns, the accusative case form (which we use for direct objects) is completely identical to the basic dictionary form.