Learn Hardcore Danish: I dag lærer vi også udtrykket "lad os", som vi bruger, når vi vil gøre noget sammen. - Today we also learn the expression “let’s,” which we use when we want to do something together.

Both are grammatically correct.
Danish main clauses must have the verb in second position (V2 rule), but the first position (the “topic”) is flexible.

I dag lærer vi også udtrykket … = I dag is the topic (what we’re talking about first: today), then the verb lærer, then the subject vi.
Vi lærer også udtrykket i dag … = now vi is the topic, then the verb lærer, and i dag is later in the sentence.

The meaning is almost the same; I dag lærer vi … puts more emphasis on today as the special time when this happens.

Where does også normally go in a Danish sentence like this? Could I say I dag vi lærer også udtrykket?

I dag vi lærer også udtrykket is not correct, because it breaks the verb‑in‑second‑position rule.
In main clauses, også typically comes right after the verb or after the subject + verb group:

I dag lærer vi også udtrykket … (verb lærer is second, også comes after subject vi)
Vi lærer også udtrykket i dag …

You usually do not put også before the finite verb in main clauses.

What exactly does lærer mean here? Is it “teach” or “learn”?

In Danish, at lære can mean both to learn and to teach, depending on the structure:

Vi lærer udtrykket = We learn the expression.
Læreren lærer os udtrykket = The teacher teaches us the expression.

In I dag lærer vi også udtrykket …, lærer clearly means “are learning”, because vi is the subject and there is no object being taught to someone else.

What is udtrykket exactly? Why does it end with -et?

Udtryk is a neuter noun (et-ord) meaning expression / phrase.

Indefinite singular: et udtryk = an expression
Definite singular: udtrykket = the expression

So udtrykket "lad os" means the expression “lad os”. The -et ending marks the definite form of a neuter noun.

What does lad os literally mean, and how is it used?

Literally, lad os is “let us”.
It is used just like English let’s to make suggestions or invitations where the speaker includes themselves:

Lad os gå. = Let’s go.
Lad os spise. = Let’s eat.

You normally follow lad os with an infinitive verb (gå, spise, tage, se etc.). It sounds neutral and is used in both spoken and written Danish.

What is lad grammatically here? Is it always an invitation word?

Lad is the imperative form of the verb at lade. At lade has several meanings, including to let / to allow and to leave.

In lad os …, it functions like English let (us) and forms the “let’s …” construction.
In other contexts, lade can mean something else:

Lad mig være. = Leave me alone.
Han lod døren stå åben. = He left the door open.

So it is not always an invitation word; it depends on the structure.

How is lad os actually pronounced in real speech?

In careful speech, you might hear something close to [ˈlað ɔs] or [ˈlað ˈɔs], but in everyday spoken Danish it often reduces:

The d can be very soft or almost disappear.
lad os can sound almost like [ˈlɒs] or [ˈlæs], depending on accent.

So Lad os gå might sound roughly like [ˈlɒs gåˀ]. Learners are usually understood even with a clearer lad os pronunciation.

Why do we use som in som vi bruger? Could we use der instead?

Both som and der can introduce relative clauses about the subject, but in this type of object relative clause (modifying udtrykket and followed by vi), som is the natural choice.

udtrykket, som vi bruger … = the expression that we use …

Der is more common when the relative pronoun is clearly the subject of the relative clause:

Manden, der står derovre, er min far. = The man who is standing over there is my father.

In som vi bruger, the subject is vi, and som is the object (“the expression that we use”), so som is preferred.

Why is there a comma before som in Danish? In English we wouldn’t always write a comma before “that/which”.

Danish comma rules traditionally require a comma before most relative clauses starting with som or der, regardless of whether they are restrictive or non‑restrictive in the English sense.

So:

udtrykket, som vi bruger, når …

Newer “comma-light” styles sometimes allow omitting that comma, but the version with the comma is still very common and always correct in standard writing. The rules are stricter than in English.

What is the difference between når and da? Why is it når vi vil gøre noget sammen?

Both can mean when, but they are used differently:

når is used for present or future time, and for repeated/general situations, including habits and rules:
- Når vi vil gøre noget sammen, bruger vi "lad os". = When(ever) we want to do something together, we use “let’s”.
da is mainly used for a single event in the past:
- Da vi ville gøre noget sammen, brugte vi "lad os". = When we wanted to do something together (that time in the past), we used “lad os”.

Here the sentence talks about a general rule, so når is correct.

Does vil in når vi vil gøre noget sammen mean “will” (future) or “want to”?

In this sentence, vil is closer to “want to” than to a pure future marker.

når vi vil gøre noget sammen = when we want to do something together

Vil often expresses intention, desire, or willingness, and only sometimes functions like a future tense. For a more explicit “want” you can also see vil gerne, but vil alone is very common in this kind of definition sentence.

Why do we say gøre noget sammen and not just gøre sammen?

In Danish, gøre (to do) almost always needs an object or a complement; gøre sammen on its own sounds incomplete.

gøre noget sammen = do something together
gøre det sammen = do it together

The noget here is a generic object (“something”), and it is necessary for the phrase to sound natural.

Why is it vi in vi vil gøre, but os in lad os? What is the difference?

Vi and os are the same pronoun but in different cases/positions:

vi = subject form (like English we)
os = object form (like English us)

So:

Vi vil gøre noget. = We want to do something. → vi is the subject.
Lad os gøre noget. = Let us do something. → os is the object of lad (“let who?” → us).

The pattern matches English: we vs us.

Could we say I dag skal vi også lære udtrykket "lad os" instead? Would that change the meaning?

Yes, you could say I dag skal vi også lære udtrykket "lad os", and it is correct.

I dag lærer vi … is more neutral: Today we are learning … / Today we learn …
I dag skal vi lære … adds a sense of plan or obligation: Today we are going to learn … / Today we have to learn …

So skal makes it sound like something scheduled or required, while lærer on its own simply states what is happening or what you are learning.