Learn Hardcore Ukrainian: Цього тижня ми читаємо іншу книгу. - This week we are reading a different book.

It literally means “of this week.” Both words are in the Genitive singular.
цього is the Genitive singular (masc./neut.) of цей (this).
тижня is the Genitive singular of тиждень (week).
Ukrainian often uses the Genitive for time expressions to mean “during/this” (adverbial Genitive), e.g., цього ранку (this morning), минулого року (last year).

Can I say на цьому тижні instead of цього тижня?

Yes. Both are natural and mean “this week.”

цього тижня = adverbial Genitive (“of this week”).
на цьому тижні = “on this week,” with the Locative: цьому (Locative of цей) + тижні (Locative of тиждень). No meaningful difference in everyday use; choose whichever feels more natural for you.

Do I have to include ми, or can I drop it?

You can drop it. Ukrainian is a pro-drop language, and the verb ending shows the subject.

With pronoun: Цього тижня ми читаємо іншу книгу.
Without pronoun: Цього тижня читаємо іншу книгу. Including ми can add emphasis or clarity, but it’s not required.

Does ми читаємо mean “we read” (habitually) or “we are reading” (right now)?

It can mean either. Ukrainian doesn’t have a separate continuous tense. Context or time words clarify:

Habit/routine: Ми читаємо багато.
Ongoing/plan: Цього тижня ми читаємо іншу книгу.
To stress “right now,” add a time adverb: Зараз ми читаємо…

Why is it іншу книгу and not інша книга?

Because книга is a direct object after читати, so it must be in the Accusative singular.

книга (Nom. fem. sg.) → книгу (Acc. fem. sg.).
The adjective agrees: інший (Nom.) → іншу (Acc. fem. sg.) to match книгу.

What exactly does іншу mean? Is it “another” or “different”? How is it different from ще одну?

іншу = “a different/other” (not the same one as before).
ще одну = “one more” (an additional one). Example:
Ми читаємо іншу книгу. = We’re reading a different book (instead of the previous one).
Ми читаємо ще одну книгу. = We’re reading one more book (in addition). Also, don’t confuse іншу with другу. In Ukrainian, другу книгу means “the second book,” not “another book.”

Can I move words around? For example, Ми цього тижня читаємо іншу книгу?

Yes. Ukrainian word order is flexible. All of these are correct, with subtle shifts in emphasis:

Цього тижня ми читаємо іншу книгу. (Time frame fronted.)
Ми цього тижня читаємо іншу книгу. (Neutral statement.)
Ми читаємо цього тижня іншу книгу. (Slight focus on the verb/action.) The most common choices are the first two.

Do I need a comma after Цього тижня?

No. A short fronted time phrase doesn’t take a comma:

Correct: Цього тижня ми читаємо іншу книгу. A comma might appear if the fronted element is long or parenthetical, but not here.

How do I turn this into a yes/no question?

Two common ways:

With the particle чи: Чи ми цього тижня читаємо іншу книгу?
Intonation only: Ми цього тижня читаємо іншу книгу? (spoken, rising intonation) Both are natural. Using чи is clear and neutral in writing.

How do I negate it, and which case should I use with інша/іншої under negation?

Basic negation: Цього тижня ми не читаємо…
If you mean “not reading any different book,” Ukrainian often uses the Genitive under negation: Цього тижня ми не читаємо іншої книги.
If it’s contrastive (“not a different one, but the same one”), keep the Accusative and add a contrast: Цього тижня ми не читаємо іншу книгу, а ту саму.

How would I say it in the past or future?

Past: Минулого тижня ми читали іншу книгу.
Future (periphrastic): Наступного тижня ми будемо читати іншу книгу.
Future (simple/synthetic): Наступного тижня ми читатимемо іншу книгу. Both future forms are standard.

Can I use книжку instead of книгу?

Yes. книжка is a common, slightly more colloquial/diminutive word for “book.”

іншу книжку (Acc. fem. sg.) is fully correct: Цього тижня ми читаємо іншу книжку.

Why does книгу end in -у? I thought only animate nouns change in the Accusative.

The animate/inanimate distinction affects mostly masculine nouns. For feminine nouns ending in -а/-я, the Accusative singular is -у/-ю regardless of animacy:

книга → книгу, газета → газету, історія → історію.

Pronunciation tips for the sentence?

цьо in цього: the soft sign ь palatalizes ц; think “ts” with a light “y” glide before “o” (approx. “tsyo”).
г in книгу is a voiced fricative [ɦ], like a breathy “h,” not an English “g.”
Distinguish vowels: і is close to English “ee” (as in іншу), while и is a more central “i” (as in ми, книгу, тижня).

Why is it г in книга/книгу and not ґ?

Ukrainian has two separate letters:

г represents [ɦ] (a voiced “h”).
ґ represents [g] and appears in relatively fewer words (e.g., ґрунт, ґаджет). The word книга historically uses г.

Can the adjective come after the noun, e.g., книгу іншу?

Standard word order is adjective + noun: іншу книгу. Postposed adjectives (книгу іншу) are rare and sound poetic or highly emphatic, so avoid them in neutral speech.

How would I say “We read a different book every week”?

Two natural options:

Щотижня ми читаємо іншу книгу.
Кожного тижня ми читаємо іншу книгу. (Here кожного and тижня are in the Genitive.)

What grammar is illustrated by each word in the sentence?

Цього: Genitive singular, masculine/neuter of цей (this).
тижня: Genitive singular of тиждень (week).
ми: 1st person plural pronoun (we).
читаємо: Present tense, 1st person plural of imperfective читати (to read).
іншу: Accusative singular, feminine of інший (other/different), agreeing with книгу.
книгу: Accusative singular, feminine of книга (book), direct object of читати.