Learn Hardcore Russian: Сначала я пишу письмо, затем читаю книгу. - First I write a letter, then I read a book.

сначала means “first” or “at first” and introduces the initial action.
затем means “then” or “after that” and introduces the following action.
Together they explicitly mark the order in which events occur.

What’s the difference between затем and потом?

Both mean “then” or “afterwards,” but затем is slightly more formal or literary, while потом is more colloquial. You can replace one with the other in most contexts, e.g. Сначала я пишу письмо, потом читаю книгу.

Why is there a comma before затем?

Here you have two independent clauses: “я пишу письмо” and “читаю книгу.” A comma is required before затем (and other similar connectors) to separate those clauses.

Why is the pronoun я included? Can it be omitted?

Russian verbs carry person and number endings, so я (“I”) is grammatically optional: Сначала пишу письмо, затем читаю книгу is perfectly normal. Including я adds emphasis or clarity.

Why are the verbs in the present tense? Isn’t this describing something in the future?

Imperfective present tense in Russian can describe habitual routines (“as a rule”), ongoing processes, or planned near-future events. In this sentence it could mean “usually I do this” or “my plan is first to write…, then to read….”

What aspect are пишу and читаю, and what would change if I used perfective verbs?

Both are imperfective, focusing on the process. If you switch to perfective—напишу and прочитаю—you shift firmly to future completed actions:
Сначала я напишу письмо, затем прочитаю книгу.

Why are письмо and книгу in the accusative case?

Both words are direct objects of their verbs. In Russian, direct objects take the accusative: neuter письмо stays письмо, feminine книга becomes книгу.

Can I change the word order for emphasis?

Yes. Russian is flexible. For example:
• Сначала я письмо пишу, затем книгу читаю. (emphasizes the objects)
• Я сначала пишу письмо, затем читаю книгу. (neutral)
Either way, the meaning stays clear because of the case endings and adverbs.