Learn Hardcore Indonesian: Kami mendaftar ke kelas lanjutan secara daring. - We register for the advanced class online.

kami excludes the listener (we but not you).
kita includes the listener (we including you). In the sentence, kami implies the listener did not register with the group.

Why use mendaftar and not daftar or mendaftarkan?

mendaftar = to register (intransitive, you register yourself).
daftar = colloquial verb for register; also a noun meaning list.
mendaftarkan [object] = to register someone/something (transitive). Examples:
Kami mendaftar ke kelas lanjutan. We registered for the advanced class.
Kami mendaftarkan adik kami ke kelas lanjutan. We registered our younger sibling for the advanced class.
Colloquial: Kami daftar kelas lanjutan.

Why is the preposition ke used? Could I use untuk, di, or pada?

ke marks the target of the registration (register to/for a class). Very common: mendaftar ke kelas/universitas.
untuk emphasizes purpose. Mendaftar untuk kelas lanjutan is also common and natural.
di/pada mark a place or platform: mendaftar di situs X, mendaftar pada panitia (formal). Less natural with a class as target. Summary: ke or untuk are both fine here; choose di/pada when focusing on where you register.

What does secara do in secara daring?

secara turns a noun/adjective into an adverb of manner. secara daring = in an online manner. Other examples: secara resmi (officially), secara otomatis (automatically).

Do I need secara before daring, or can I just say daring or online?

Standard/formal when modifying a verb: mendaftar secara daring or mendaftar secara online.
Colloquial: daftar online (very common).
daring by itself is more often used attributively: kelas daring, pembelajaran daring. As an adverb without secara, it’s less common in neutral writing.

Does secara daring describe the class as online, or the act of registering?

It modifies the verb mendaftar, so it says the registration was done online. It does not say whether the class itself is online.

How do I say that the class itself is online?

Use an attributive form:

kelas lanjutan daring
kelas lanjutan online
More explicit: kelas lanjutan yang diadakan/diselenggarakan secara daring

Can I move secara daring elsewhere in the sentence?

Yes. Common options:

Kami mendaftar ke kelas lanjutan secara daring. (original)
Kami mendaftar secara daring ke kelas lanjutan.
Secara daring, kami mendaftar ke kelas lanjutan. (more formal/emphatic at the start)

How do I mark past, present, or future since Indonesian has no tense?

Use particles or time words:

Past/completed: Kami sudah/telah mendaftar...
Ongoing: Kami sedang mendaftar...
Future/intention: Kami akan/mau mendaftar...
Recent: Kami baru saja mendaftar...

Can I drop the subject kami?

Yes, if context makes it clear:

(Kami) mendaftar ke kelas lanjutan secara daring. Indonesian often omits pronouns when they’re inferable.

What’s the nuance of lanjutan versus lanjut, tingkat lanjut, or mahir?

lanjutan = continuation/advanced level (very common in course names): kelas lanjutan.
lanjut = further/continue; less used as a course label.
tingkat lanjut = advanced level (more formal/plannish): kelas tingkat lanjut.
mahir = proficient/skillful; suggests high proficiency: kelas mahir.

Is kelas the same as kursus or mata kuliah?

kelas = a class or class group.
kursus = a course, often non-formal training (e.g., language course).
mata kuliah = a university course/subject. So you could have kelas within a kursus or a mata kuliah.

How would this sound in casual speech?

Natural casual variants:

Kita daftar kelas lanjutan online. (listener included)
Kami daftar online buat kelas lanjutan. Dropping men- and using online and buat is common in conversation.

How do I make it more formal or passive?

Passive/impersonal: Pendaftaran ke kelas lanjutan dilakukan secara daring.
Formal active with a noun: Kami melakukan pendaftaran ke kelas lanjutan secara daring.
Passive with an agent: Kami didaftarkan ke kelas lanjutan secara daring (oleh panitia).

Is mendaftar kelas lanjutan okay without a preposition?

In careful/standard Indonesian, prefer mendaftar ke/untuk kelas lanjutan. In casual speech you will hear daftar kelas lanjutan, but formal writing expects a preposition.

What does daring literally mean, and what’s the opposite?

daring comes from dalam jaringan (in-network) = online. The counterpart is luring (luar jaringan) = offline. Examples: kelas daring (online class), kelas luring (offline class).

Why is it mendaftar (with meN-) morphologically?

The prefix meN- forms active verbs. With a base starting in d (daftar), it surfaces as men-, giving mendaftar. This is the standard active intransitive form meaning to register.