Learn Hardcore Ukrainian: Тобі варто виходити раніше, щоб встигати без поспіху. - You should leave earlier in order to make it without rushing.

Because варто is an impersonal predicate meaning “it’s advisable/it’s worth [doing].” Ukrainian uses the dative to mark the person to whom the recommendation applies (the “dative of experiencer”): Тобі варто… = “For you, it’s advisable to…”.
Other dative pronouns you’ll hear:

Мені варто… (for me)
Йому/їй варто… (for him/her)
Нам/вам варто… (for us/you-pl/polite)
Їм варто… (for them)

What’s the nuance of варто compared with треба / потрібно / слід / мусиш / повинен?

варто: soft, polite recommendation (“you’d do well to…”).
треба / потрібно: neutral necessity (“it’s necessary/you need to…”).
слід: “ought to,” somewhat formal/bookish.
мусиш / повинен: strong obligation (“you must/you’re obliged to”). Your sentence is advice, not an obligation, so варто fits well.

Does варто always need a person in the dative?

No. You can give general advice with no person: Варто виходити раніше… (“It’s advisable to leave earlier…”). Adding a dative (“Тобі варто…”) personalizes who the advice is for.

Why is the verb виходити (imperfective) used here instead of вийти (perfective)?

Because the sentence gives a general, habitual recommendation. Imperfective виходити suits habits and repeated actions. If you were talking about a one-time plan, you’d switch to perfective:

Тобі варто вийти раніше, щоб встигнути без поспіху. (“You should leave earlier [this time] to make it without rushing.”)

What’s the difference between встигати and встигнути here?

встигати (imperfective): “to manage/to be on time” habitually or generally.
встигнути (perfective): “to manage/make it” once or as a completed result.
So your sentence advises a habit: leave earlier so you habitually make it without rushing. For a single event, use встигнути.

What does щоб do in this sentence, and what can come after it?

щоб introduces a purpose clause (“so that/in order to”). When the subject is the same in both parts, Ukrainian commonly uses an infinitive: …щоб встигати. Alternatives:

Infinitive, perfective for a single result: …щоб встигнути.
A finite clause if you name a different subject: …щоб він встиг.

Is the comma before щоб necessary?

Yes. Ukrainian separates the main clause from a subordinate purpose clause with a comma: …, щоб ….

What case is поспіху after без, and could I say this another way?

без requires the genitive, so поспіху is genitive singular. Natural alternatives:

…щоб не поспішаючи. (using a verbal adverb, “without hurrying”)
…щоб спокійно/неквапно. (adverbs: “calmly/unhurriedly”)

Can I change the word order?

Yes, word order is flexible for nuance:

Тобі варто виходити раніше, щоб встигати без поспіху. (neutral)
Варто тобі виходити раніше… (slight emphasis on the recommendation)
…щоб без поспіху встигати. (stylistic reshuffle; meaning unchanged)

What exactly does раніше modify, and how “early” is it?

раніше is the comparative adverb of рано (“early”) and modifies виходити: leave earlier than you currently do or earlier than some implicit reference time. You can grade it: трохи раніше (a bit earlier), значно раніше (much earlier).

How do I say this politely to someone I don’t know well?

Use polite dative plural вам and plural imperative if needed:

Вам варто виходити раніше, щоб встигати без поспіху.
Imperative version: Виходьте, будь ласка, раніше, щоб встигати без поспіху.

How would I turn this into a direct command?

Use the imperative:

Виходь раніше, щоб встигати без поспіху. (singular, informal)
Виходьте раніше, щоб встигати без поспіху. (plural/polite)

How do I say the opposite—“you shouldn’t”?

Add negation to the impersonal predicate:

Тобі не варто виходити так пізно. (“You shouldn’t leave so late.”)

Are there close synonyms for the purpose part?

Yes, for a more direct “avoid being late,” you can say:

…щоб не запізнитися. (perfective, one-time)
…щоб не спізнюватися. (imperfective, habitually) These focus on not being late rather than on “without rushing.”

How is this pronounced? Where are the stresses?

Typical stresses:

Тобі́ ва́рто виходи́ти рані́ше, щоб встига́ти без по́спіху. This will sound natural across regions.