Learn Hardcore Japanese: watasi ha nikki ni eiga no kansou wo kakimasu. - I write my impressions of the movie in my diary.

は marks the topic of the sentence: what we’re talking about.
So 私は means roughly “As for me,” or “Speaking about me,”.
が marks the grammatical subject, especially when introducing new information or contrasting who actually did something.
Here, the speaker is just setting themselves up as the topic (we’re talking about what I do), so 私は is natural. Using 私が would sound more like you’re contrasting with someone else: “I (not someone else) will write my impressions in my diary.”

Can I drop 私 and just say 日記に映画の感想を書きます?

Yes. In natural Japanese, the subject (私) is often omitted when it’s clear from context.
日記に映画の感想を書きます will normally be understood as “I write / will write my impressions of the movie in (my) diary.”
You keep 私は if you particularly want to emphasize “me” or contrast yourself with others, or if it’s the first time you’re clearly establishing who the subject is.

What exactly does 日記に mean? Why use に here, not で?

Here に marks the target / destination of the writing: the diary is the place where the writing ends up.
So 日記に書きます = “write in the diary / into the diary.”
で usually marks the place where an action occurs (e.g. 図書館で勉強します – “I study at the library”), but when you write onto/into something (paper, a board, a diary, the wall), Japanese normally uses に, because that thing is treated as the target of the writing:

ノートに書く – write in a notebook
黒板に書く – write on the blackboard
日記に書く – write in a diary

In English I’d say “write about the movie.” Why is 感想 (impressions) the direct object, not 映画?

Japanese focuses on what physically gets written, so the direct object is 感想 (what you actually put into words).
The movie is the topic or source of those impressions, not the thing you literally write.
So:

映画の感想を書く = “to write (one’s) impressions of the movie”
If you said 映画を書く, that would sound like “to write (create/compose) a movie (script),” which is different.

What is the function of の in 映画の感想?

の here links two nouns and is similar to English “of” or sometimes a possessive “’s”.
映画の感想 literally means “impressions of the movie” or “the movie’s impressions.”
More generally, XのY can mean many relationships: possession, “about,” “related to,” etc. In this case it’s “impressions about the movie.”

Why does 感想 take を? Could it take は instead?

を marks the direct object of the verb. 感想を tells us what is being written: “write impressions.”
You can say 映画の感想は日記に書きます, but now 映画の感想 is the topic (marked by は), not the direct object marker. It’s like saying:

“As for my impressions of the movie, (I) write them in my diary.”
Grammatically, in that version the direct object is just understood/omitted, but the meaning is almost the same. The original sentence with を is more straightforward: “I write my impressions (of the movie) in my diary.”

Can I change the word order, like 映画の感想を日記に書きます?

Yes. Japanese word order is fairly flexible as long as the verb comes at the end and particles stay with the words they mark.
All of these are grammatical and natural:

私は日記に映画の感想を書きます。
私は映画の感想を日記に書きます。
映画の感想を日記に書きます。 (dropping 私)
The most neutral/common order here is to put を-marked and に-marked phrases before the verb in either order. Changing the order slightly affects focus, but not the basic meaning.

Does 書きます mean “write” (present) or “will write” (future)?

書きます is non-past; it covers both present and future.
Context decides whether it’s understood as:

habitual / general present: “I (usually) write my impressions of movies in my diary.”
future / planned action: “I will write my impressions of the movie in my diary (later).”
Japanese doesn’t have a separate future tense; the same form is used, and time is inferred from context or added time expressions.

What is the difference between 書きます and 書く?

Both mean “to write”; the difference is politeness level.

書きます is the polite -ます form, used in most formal or neutral situations (with strangers, at work, in writing, etc.).
書く is the plain form, used in casual speech (with friends, family, diaries, fiction narrative, etc.).
So:
日記に映画の感想を書きます。 – polite
日記に映画の感想を書く。 – casual/plain

Why don’t we say 私の日記に? How do listeners know it’s my diary?

Japanese often omits obvious possessors like “my,” “your,” “our.”
In this context, if you say 日記に映画の感想を書きます, it’s naturally understood as “write in my diary”, because people normally write in their own diaries.
You can say 私の日記に映画の感想を書きます to emphasize that it’s my diary (for example, contrasting with someone else’s diary), but in everyday speech 私の is usually left out if the owner is clear from context.