Learn Hardcore Japanese: watasi ha kyuuzitu ni hiroi sibahu de hirune wo simasu. - On my day off, I take a nap on a wide lawn.

は after 私 marks the topic: “As for me…”
に after 休日 marks the time: “on (my) days off/holidays.”
で after 芝生 marks the place where the action happens: “on/at the lawn.”
を marks the direct object of the verb する; here the object is the verbal noun 昼寝 “nap.”
します is the polite non-past form of する (“do”), so the sentence is polite: “I take naps…”

Why is を used with 昼寝? Can I just say 昼寝します?

昼寝 is a verbal noun (a “suru-verb” noun). With these, both forms are common:

With object marker: 昼寝をします
Without it: 昼寝します Meaning is the same. Omitting を is slightly snappier/casual; including を can sound a touch more careful, but both are perfectly natural.

Why is it 芝生で and not 芝生に?

で marks the location where an action is performed: 芝生で昼寝する = “nap on the lawn.”
に marks a destination or a location of existence. You might see 芝生に寝る (“to lie down on the lawn,” focusing on the target you lie on), but for the ongoing action of napping, で is the normal choice.
Analogy: 図書館で勉強する (study at the library) vs 図書館に行く (go to the library).

Do I need to say 私? Don’t Japanese sentences often omit the subject?

You can omit 私 if it’s clear who the subject is:

休日に広い芝生で昼寝をします。 (Natural and common) Use 私 when you need to introduce or contrast the topic (“As for me…”), or where context is unclear.

What’s the difference between 休日, 休み, and 休みの日? Could I use 週末?

休日 (きゅうじつ): “a day off/holiday” (more formal/official-sounding; e.g., company-designated days off).
休み (やすみ): time off, break, vacation; very common and casual but can be vague.
休みの日: “a day that I’m off” — idiomatic and clear in everyday speech.
週末: “the weekend” specifically.
All can fit, depending on your exact meaning:
Official day off: 休日に…
Just “on my days off”: 休みの日に…
Specifically weekends: 週末に…

Should it be 芝生の上で instead of 芝生で to mean “on the grass”?

Both are fine:

芝生で is natural and generally understood as “on the lawn.”
芝生の上で explicitly says “on top of the lawn/grass” and sounds a bit more descriptive or careful.
Use の上で when you want to emphasize the surface.

How does the adjective 広い work here? Could I say 広々とした芝生?

広い is an -i adjective directly modifying 芝生: “a wide/spacious lawn.”
広々とした芝生 is also natural and can sound a bit more vivid (“a very spacious lawn”).
Both are good; choose based on nuance.

Can I change the word order? And what about 休日には?

Japanese allows flexible order before the verb. A common flow is Topic → Time → Place → Object → Verb:

Natural: （私は）休日に広い芝生で昼寝をします。
Also possible: 休日には、（私は）広い芝生で昼寝をします。
Using には topicalizes and emphasizes the time phrase (often with a contrastive feel: “On my days off (as opposed to other times), …”). Putting time first is typical; putting place first is also okay if you want to highlight it.

Why use the polite します instead of する? What other forms might I need?

Can any particles be dropped in casual speech?

Often dropped: 私, and を with する-nouns: 昼寝する.
Sometimes dropped: time に in casual flow, but with nouns like 休日 keeping に is clearer.
Keep: で for place of action; dropping it risks ambiguity.
Example casual: 休日、広い芝生で昼寝する。

Should it be 私は or 私が?

私は sets “me” as the topic; neutral and most common here.
私が marks “I” as the subject and often emphasizes “I (as opposed to others).”
Example:
Q: 誰が休日に広い芝生で昼寝をしますか。
A: 私が（します）。
In a neutral statement like yours, 私は (or omitting 私) is standard.

How do you pronounce each part?

私: わたし (watashi)
Particle は: pronounced “wa”
休日: きゅうじつ (kyuujitsu)
に: ni
広い: ひろい (hiroi)
芝生: しばふ (shibafu)
で: de
昼寝: ひるね (hirune)
Particle を: pronounced “o”
します: します (shimasu)
Full reading: わたしはきゅうじつにひろいしばふでひるねをします。
Romaji: Watashi wa kyuujitsu ni hiroi shibafu de hirune o shimasu.