Learn Hardcore Norwegian: Jeg henter babyen i barnehagen etter jobb. - I pick up the baby at daycare after work.

habitual actions: Jeg henter babyen … = “I pick up the baby … (as a routine)”
scheduled/near-future plans, especially with time context: … etter jobb makes it naturally read as “I’m picking up the baby after work (today/then).”

If you want to sound more explicitly future/intended, you can say:

Jeg skal hente babyen i barnehagen etter jobb. (very common)
Jeg skal hente babyen etter jobb.

What’s the difference between å hente and å ta here?

Å hente means to go and get someone/something and bring them back (or collect them), which is exactly what you do with a child at daycare.
Å ta is broader (take) and can sound vague or odd in this context unless expanded:

Jeg henter babyen … = I collect/pick up the baby.
Jeg tar babyen … alone can sound like “I take the baby” (not specifically “pick up from daycare”).

Why is it babyen and not en baby?

Babyen is the definite form: the baby. In real-life contexts (your own baby, the specific baby you’re talking about), Norwegian typically uses the definite form:

en baby = “a baby” (any baby, introducing it for the first time, or not specific)
babyen = “the baby” (the specific one you mean)

How do I know the gender here—could it be babya or babyet?

Baby is commonly treated as masculine: en baby – babyen. In some varieties you may see feminine (ei baby – babya) if the speaker uses feminine forms actively. Neuter (et baby) is less common.

So the safest, most standard learner choice is:

en baby – babyen

Why is it i barnehagen and not på barnehagen?

With places you can enter/be inside, Norwegian often uses i = in/at (inside):

i barnehagen suggests the baby is picked up at/in the daycare building/facility.

På is more like on/at (a site/area/activity) and can be used in some “institution” contexts, but i barnehagen is the most straightforward here.

Why does barnehagen have -en at the end?

That’s the definite ending for a masculine noun:

en barnehage = a daycare/kindergarten
barnehagen = the daycare (the specific one)

Norwegian often attaches the to the end of the noun rather than using a separate word.

Is barnehage the same as “kindergarten,” “nursery,” or “daycare”?

Barnehage is the general Norwegian term for daycare / preschool (roughly ages 1–5/6). It can overlap with what English speakers call nursery or preschool; it’s not exactly the same as “kindergarten” in every school system, but it’s often translated that way depending on context.

Why is it etter jobb and not etter jobben?

Both can be grammatical, but they differ in feel:

etter jobb = “after work” (after the workday; a set phrase-like, general time reference)
etter jobben = “after the job/work” (more specific/definite; can sound a bit heavier or more concrete)

In everyday Norwegian, etter jobb is very common for “after work.”

Where does i barnehagen go in the sentence? Could it move?

Yes, Norwegian word order is flexible with adverbials (time/place phrases). All of these are natural with slightly different emphasis:

Jeg henter babyen i barnehagen etter jobb. (neutral)
Jeg henter babyen etter jobb i barnehagen. (can sound a bit “stacked,” but still possible)
Etter jobb henter jeg babyen i barnehagen. (emphasizes time; note verb-second rule: henter stays in 2nd position)

Do I need på with jobb (like på jobb)?

Not here. Etter jobb is the standard way to say “after work.”
På jobb means “at work / at the workplace,” and it would be used in different sentences:

Jeg henter babyen når jeg er ferdig på jobb. = when I’m done at work
Jeg er på jobb nå. = I’m at work now