Learn Hardcore Spanish (Spain): ¿Cuánto te cuesta la matrícula este semestre? - How much does the tuition cost you this semester?

Te is an indirect object pronoun meaning to you / for you.
In Spanish, costar often works like: Something costs (to someone) X.
So structurally it’s: [La matrícula] (subject) + [te] (to you) + [cuesta] (costs).

Why is it cuesta (singular) and not cuestan?

Because the grammatical subject is la matrícula (singular), so the verb agrees with it: la matrícula cuesta.
You’d use cuestan if the subject were plural, e.g. Las matrículas cuestan mucho.

What is the base verb here, and is it irregular?

The verb is costar (to cost). It’s a stem-changing verb: o → ue in most present tense forms:

cuesta, cuestan
cuesto, cuestas, cuestan (etc.)

Why is it ¿Cuánto…? with an accent, but sometimes I see cuanto without one?

With an accent (cuánto) it’s interrogative/exclamatory: how much / how many.
Without an accent (cuanto) it’s usually relative: as much as / whatever amount.
Examples:

¿Cuánto cuesta? (question)
Te pago cuanto me pidas. (whatever amount you ask)

Is la matrícula the same as tuition? I’ve heard it can mean other things.

In Spain, la matrícula commonly means enrolment/registration fee and often by extension the tuition payment for a term. Context (university, school) makes it clear.
It can also mean license plate (also common in Spain), but not in this context.

Why does Spanish use la here? Can I omit the article?

Spanish often uses the definite article (el/la/los/las) in places where English doesn’t. La matrícula is natural here because it refers to the enrolment/tuition fee (for that term).
You can omit it in some contexts, but it can sound less specific or more like a general concept: ¿Cuánto cuesta matrícula…? is much less standard than ¿Cuánto cuesta la matrícula…?

What’s the difference between ¿Cuánto cuesta la matrícula…? and ¿Cuánto te cuesta la matrícula…?

Both are correct. Adding te highlights that it’s your cost (what you personally have to pay), which can matter if different students pay different amounts (scholarships, residency status, etc.).

¿Cuánto cuesta la matrícula este semestre? = general price
¿Cuánto te cuesta…? = your price

How would I say this formally (to someone I’d address as usted)?

Change te to le:
¿Cuánto le cuesta la matrícula este semestre?
(And if you want to be extra clear, you can add a usted: ¿Cuánto le cuesta a usted…?)

Why is the word order like this? Could it be ¿Cuánto cuesta la matrícula este semestre?

Yes, that alternative is completely natural. Spanish allows flexibility, especially with pronouns. Common options include:

¿Cuánto te cuesta la matrícula este semestre?
¿Cuánto cuesta la matrícula este semestre?
The pronoun te typically goes before the conjugated verb (te cuesta).

Is este semestre the usual way to talk about academic terms in Spain?

It’s understood and correct. In Spain, you’ll also often hear este cuatrimestre (many universities divide the year into two cuatrimestres rather than “semesters” in everyday admin language). Usage depends on the institution.

Could I replace costar with valer here?

Sometimes, but there’s a nuance. Valer is more like to be worth / to cost (price tag) and is common in casual price questions: ¿Cuánto vale?
For fees and what you pay, costar is very common and often sounds more appropriate: ¿Cuánto cuesta la matrícula…?

If I wanted to emphasize “to you (not to someone else)”, how would I do that?

You can add the prepositional phrase a ti for emphasis:
¿Cuánto te cuesta a ti la matrícula este semestre?
This is used when contrasting with someone else’s cost.