Learn Hardcore Dutch: Overmorgen ga ik stage lopen bij de bibliotheek. - The day after tomorrow I am going to do an internship at the library.

Two words (over morgen) would mean “about tomorrow” (as in talking about tomorrow), which is different.
Related words: eergisteren = the day before yesterday; colloquial overovermorgen = the day after the day after tomorrow.

Why is it Overmorgen ga ik, not “Overmorgen ik ga”?

Dutch main clauses are verb-second (V2). Whatever you put first (here: Overmorgen), the finite verb must come second. So:

Correct: Overmorgen ga ik stage lopen ...
Also correct if the subject comes first: Ik ga overmorgen stage lopen ...

What does gaan + infinitive (here: ga … lopen) express?

It’s the usual way to express a planned or intended near future, much like English “going to.” It doesn’t emphasize physical movement; it signals intention/schedule:

Ik ga stage lopen = I’m going to do an internship.

Could I use zal instead of ga?

Grammatically yes: Overmorgen zal ik stage lopen ... But in everyday speech gaan is more natural for plans. Zullen often sounds more formal or predictive (offers, promises, assumptions), not so much like an already arranged plan.

Why stage lopen? Does lopen really mean “walk” here?

No. Stage lopen is a set expression meaning “to do an internship.” Lopen is a “light verb” here. Alternatives:

Stage lopen: most idiomatic and common.
Stage doen: also very common and fine in everyday language.
(Mijn) stage lopen: e.g., Ik loop mijn stage bij … emphasizes “my internship.”
Stage volgen: more formal/educational-register (“follow a placement/course”). All mean essentially the same in typical contexts.

Can I say een stage or must it be bare stage?

Both occur. Use the bare form for the general activity; use een stage when you mean “one internship” or you specify details:

General: Ik loop (ga) stage lopen.
Countable: Ik loop een stage van drie maanden bij de bibliotheek.

How do I negate this correctly: niet or geen?

To say you’re not doing an internship at all: use geen (negates the noun stage):
- Overmorgen ga ik geen stage lopen (bij de bibliotheek).
To say you will do an internship, but not at the library: use niet with the place phrase:
- Overmorgen ga ik niet bij de bibliotheek stage lopen.

Why bij de bibliotheek and not in/voor/naar?

bij
- organization = working at/with that organization: stage lopen bij de bibliotheek.
in = physically inside: in de bibliotheek studeren (to study in the library).
voor = on behalf of/for the benefit of: werken voor de bibliotheek.
naar = motion towards: Ik ga naar de bibliotheek (I’m going to the library).

Do I need the article de in bij de bibliotheek?

Yes, because bibliotheek is a common noun and a de-word: de bibliotheek. With company names you usually drop the article: bij Google, bij Albert Heijn. With institutions as common nouns: bij de universiteit, bij het ziekenhuis, bij de bibliotheek.

Can I change the word order?

Yes, within the usual Dutch patterns. All of these are fine (with slightly different emphasis):

Overmorgen ga ik stage lopen bij de bibliotheek. (given sentence)
Ik ga overmorgen stage lopen bij de bibliotheek. (subject first)
Overmorgen loop ik stage bij de bibliotheek. (simple present used for scheduled events)
Overmorgen ga ik bij de bibliotheek stage lopen. (puts the place right before the verb phrase) Keep the finite verb second in main clauses.

How would this look in a subordinate clause?

In a “dat”-clause, the verbs go to the end:

Ik denk dat ik overmorgen bij de bibliotheek stage ga lopen. Here the non-finite cluster is at the end; placing stage between ga and lopen is natural.

Does the sentence imply I’m starting the internship that day?

Not necessarily. It means you’ll be doing internship work that day. If you specifically mean “start,” say:

Overmorgen begin ik (aan/met) mijn stage bij de bibliotheek.

Is bibliotheek “bookstore”?

No. bibliotheek = library (borrow books). Bookstore is boekhandel or boekwinkel.

How do I pronounce the tricky parts?

g in ga is a guttural fricative ([ɣ] or , depending on region).
stage = [ˈstaːʒə]; the g-sound is like the “zh” in “measure.”
bij = [bɛi], roughly like English “eye.”
bibliotheek ≈ [ˌbiblioːˈteːk]; stress on the last part, long “ee” in -teek. A full slow IPA rendering: [ˈoːvərˌmɔrɣə(n) ɣaː ɪk ˈstaːʒə ˈloːpə(n) bɛi də ˌbiblioːˈteːk].