Learn Hardcore Esperanto: Sabato estas trankvila tago por mia familio.

Yes. In Esperanto, names of the days of the week (like Lundo, Mardo, Sabato) are written with a capital letter, just like proper names.
So Sabato is capitalized in the same way that Saturday is in English.

Why is there no la before Sabato? Shouldn’t it be La sabato estas trankvila tago…?

You leave out la here because Sabato is used in a general sense: “Saturday (as a day in general) is a quiet day for my family.”

Sabato estas trankvila tago… ≈ “Saturday is a quiet day…” (speaking generally)
La sabato estas trankvila tago… would sound more like “The Saturday is a quiet day…”, referring to a specific Saturday, which is not the intention here.

Also, Esperanto has only the definite article la; there is no separate word for “a/an”.

Does Sabato estas trankvila tago mean “Saturday is a quiet day” or “Saturday is the quiet day”?

The most natural English translation here is “Saturday is a quiet day.”

In Esperanto:

The lack of la in trankvila tago makes it indefinite in feel: more like “a quiet day” than “the quiet day.”
If you really needed “the quiet day,” you could say la trankvila tago, but in this kind of general statement, that would be unusual.

Why does trankvila come before tago? Can I also say tago trankvila?

Both orders are grammatically correct in Esperanto:

trankvila tago – a quiet day
tago trankvila – also “a quiet day,” but with a slightly more stylistic or poetic feel.

The neutral, most common order is adjective + noun (trankvila tago). Noun + adjective is often used to emphasize or for stylistic effect.

Why does trankvila end in -a and tago in -o?

In Esperanto, word endings show the part of speech:

-o = noun → tago (“day”), familio (“family”)
-a = adjective → trankvila (“quiet, calm”), mia (“my”)

So trankvila tago literally is “quiet (adj.) day (noun).”
The endings -a and -o must agree: an adjective describing a noun will have -a, the noun will have -o.

Why doesn’t trankvila tago (or any word here) have an -n at the end? When should I use the accusative -n?

The accusative -n is mainly used for:

Direct objects of verbs:
- Mi legas libron. – I am reading a book. (libron = object)
Showing direction of movement (with some prepositions):
- Mi iras en la domon. – I go into the house.

In Sabato estas trankvila tago por mia familio:

There is no direct object; Sabato and trankvila tago are joined by estas (a linking verb).
por mia familio uses por, a preposition that does not require the accusative for direction.

So everything correctly stays without -n.

What exactly does por mean here? Why por mia familio and not al mia familio?

Por generally means “for” in the sense of “intended for,” “benefiting,” or “in relation to.”

por mia familio = “for my family” (from my family’s perspective / in my family’s case).

Al usually means “to” and normally expresses movement or giving something to someone:

Mi donas libron al mia familio. – I give a book to my family.

In your sentence you’re not giving anything to the family; you’re describing how Saturday is for them, so por is correct.

Why is it mia familio and not la mia familio?

In Esperanto, possessive adjectives like mia, via, lia, ŝia, nia, ilia usually replace la, rather than combine with it:

mia familio = my family (not “the my family”)
via libro = your book

You would normally not say la mia familio. Forms like la mia can exist, but only in specific structures, e.g.:

Tiu libro estas la mia. – That book is mine.

In front of a noun, just use mia familio.

Why is familio singular, even though “my family” is a group of people?

In Esperanto, familio is a singular collective noun, just like “family” in standard English grammar:

Mia familio estas granda. – My family is big.

If you want to refer explicitly to the individual family members, you’d make a different noun and use the plural:

familiano = family member
familianoj = family members
Sabato estas trankvila tago por la familianoj. – Saturday is a quiet day for the family members.

But in your original sentence, the collective singular familio is natural and correct.

How would I say “Saturdays are quiet days for my family” (plural) in Esperanto?

You would pluralize both the subject and the predicate noun and adjective:

Sabatoj estas trankvilaj tagoj por mia familio.

Breakdown:

Sabatoj – Saturdays
estas – are
trankvilaj tagoj – quiet days
por mia familio – for my family

Notice how both the noun (tagoj) and the adjective (trankvilaj) take -j in the plural.

Can I leave out estas in this kind of sentence, like some languages do?

No. In standard Esperanto, the verb esti (“to be”) is not normally omitted.
You need estas here:

Sabato estas trankvila tago… – correct
Sabato trankvila tago… – incorrect / ungrammatical in normal Esperanto.

Omitting estas is not a feature of ordinary Esperanto syntax.

How is this sentence pronounced and where is the stress?

In Esperanto, stress is always on the second-to-last syllable of a word. A simple breakdown:

Sabato → sa-BA-to
estas → ES-tas
trankvila → tran-KVI-la
tago → TA-go
por → (one syllable, por)
mia → MI-a
familio → fa-mi-LI-o

So the whole sentence is:
sa-BA-to ES-tas tran-KVI-la TA-go por MI-a fa-mi-LI-o.