Learn Hardcore Chinese: jīntiān de gōngzuò hěn zhòngyào.

的 marks an attributive relationship. 今天的工作 literally means "today’s work" (the work belonging to or scheduled for today). Here 今天 modifies 工作 via 的.

Can I omit 的 and say 今天工作很重要? What’s the difference?

Yes, but the nuance changes:

今天的工作很重要 = "Today’s work (the tasks for today) is important."
今天工作很重要 = "Today, work is important" (今天 is a time word for the whole sentence; more about the day than about a specific set of tasks).

Why is 很 there? Does it always mean "very"?

In simple statements with an adjective predicate, 很 often works as a neutral "linking" degree adverb so the sentence sounds natural. It doesn’t necessarily mean "very" here; it can just mean "is." To strongly mean "very," use 非常/特别/真的很.

Can I drop 很 and say 今天的工作重要?

Grammatically yes, but it sounds contrastive or emphatic, like "Today’s work (as opposed to something else) is important." For a neutral statement, 今天的工作很重要 is more natural.

Where is the verb "to be"? Why no 是?

Chinese adjectives can function as predicates (stative verbs). You typically don’t use 是 before an adjective. So 工作很重要 is the normal pattern, not 工作是重要.

Can/should I use 是, like 今天的工作是重要的 or 是很重要(的)?

今天的工作是重要的: acceptable, more formal/emphatic (the adjective is nominalized by 的).
今天的工作是很重要的: often used to concede/emphasize ("It really is very important").
Neutral description: prefer 今天的工作很重要.
Avoid …是重要 (without sentence-final 的) in modern speech; it sounds incomplete.

How do I ask this as a yes–no question?

今天的工作重要吗？ (natural; here you don’t need 很)
今天的工作重不重要？ (A–not–A form)
今天的工作很重要吗？ specifically asks "Is it very important?"

How do I negate it naturally?

今天的工作不重要 = not important.
今天的工作不太重要 = not very important (softer).
今天的工作不是很重要 = not particularly important (very common). Use 不 (not 没) to negate adjectives like 重要.

What are alternatives to 很 for degree?

Stronger: 非常/特别/格外/相当 重要
Colloquial: 挺重要
Exclamatory: 太重要了 Choose based on tone and formality.

What’s the tone/pronunciation of 重 in 重要?

In 重要, 重 is zhòng (4th tone), meaning "heavy/important." The same character is zhōng (1st tone) in words like 中国 (Zhōngguó). 重要 = zhòngyào (both 4th tone). 很 is 3rd tone; here it’s usually a low "half-third."

Do any tone changes (sandhi) happen with 很 here?

Not a major one. The general rule: a 3rd tone before another 3rd becomes rising (2nd), e.g., 很好 → hén hǎo. Here, 很 (3rd) is followed by 重 (4th), so no 3rd–3rd sandhi applies.

Is 工作 a noun or a verb here? Do I need a measure word?

Here 工作 is a noun ("work/tasks"), so no measure word is needed. For a specific job/assignment, use a classifier:

这份工作 (this job)
一项重要的工作 (an important task)

What’s the difference between 重要, 要紧, and 关键?

重要: neutral "important."
要紧: colloquial "urgent/important."
关键: "crucial/key," decisive to an outcome. Choose based on nuance and register.

Can I put the adjective before 工作, like "important work today"?

Yes, as a modifier with 的:

今天有重要的工作 = "There is important work today." Predicate position uses no 的: 工作很重要.

How would I answer the question 今天的工作怎么样?

很重要。
不太重要。
非常/特别重要。 Dropping the subject is natural in replies.

Can I use 今日 instead of 今天?

Yes. 今日 is more formal/literary: 今日的工作很重要. In everyday speech, 今天 is standard.

Where can time words go here?

Inside the subject via 的: 今天的工作很重要.
As a sentence-level time word: 今天，工作很重要. Avoid 工作今天很重要 in this meaning; it’s awkward here.

Is sentence-final 的 possible: 今天的工作很重要的?

Yes. Sentence-final 的 adds a soft, insistent tone: "Today’s work really is important (you know)." It’s common in spoken Chinese for reassurance or emphasis.