Learn Hardcore Malay: Selepas mengisi borang, saya terus ke kaunter bagasi. - After filling in the form, I go straight to the luggage counter.

In Malay, meN- is an active‐verb prefix. When you attach it to the root isi, you get mengisi (“to fill in”). Without the prefix, isi is either:

A noun meaning “contents” (“isi buah” = fruit flesh)
An imperative “Fill!”
So mengisi borang is the correct verb phrase for “filling out the form.” Phonologically, meN- becomes meng- before a vowel (i), hence meng+isi = mengisi.

What does selepas mean, and how is it different from setelah or just lepas?

All three can mean “after,” but with different registers:

selepas: Standard, neutral (“Selepas hujan, kami keluar.”)
setelah: Formal or literary (“Setelah selesai, dia pulang.”)
lepas: Colloquial/informal (“Lepas makan, kita pergi.”)
In your sentence, selepas is the everyday written choice for “after.”

Can I move selepas mengisi borang to the end of the sentence? How would that sound?

Yes. Malay allows flexible clause order. You could say:
Saya terus ke kaunter bagasi selepas mengisi borang.
Both versions emphasize the same sequence; placing the time clause first adds a slight focus on what happened first.

What does terus mean in saya terus ke kaunter bagasi? Could I use teruskan or langsung instead?

Here terus is an adverb meaning “straightaway” or “immediately” (no delay).

langsung can sometimes replace terus (“saya langsung ke sana”), though langsung also carries “direct/don’t detour.”
teruskan is a verb meaning “continue something” (e.g., teruskan kerja = “continue working”), so it wouldn’t make sense here.

Why is there no untuk before mengisi borang in selepas mengisi borang?

Selepas is a conjunction that directly links to a clause (mengisi borang). Adding untuk would change the meaning to “in order to fill the form” rather than “after filling the form.” In time‐sequence clauses, you don’t need untuk.

Is saya necessary here? Can I replace it with another pronoun or omit it?

saya is the first‐person formal pronoun “I.”
You could replace it with aku (informal) or kami/kita (we) if context changes.
Malay often drops pronouns when context is clear, but starting with saya makes the subject explicit and polite.

Why is it ke kaunter bagasi and not di kaunter bagasi?

ke indicates motion toward a place: “to the baggage counter.”
di indicates location or position: “at the baggage counter.”
Since you’re going there, ke is the correct preposition.

What exactly does borang refer to? Can it be used in other contexts?

Borang means a (usually printed or digital) “form” you fill in with information. Common collocations:

borang pendaftaran (registration form)
borang cukai (tax form)
You wouldn’t use borang for generic paper; it’s specifically a document designed to be completed.