Learn Hardcore Swedish: Den här gången måste vi vara mer organiserade.

In Swedish, den här goes with utrum (common‐gender) nouns and det här with neuter‐gender (ett) nouns. Gången is common‐gender (en‐word), so you say den här gången (“this time/occasion”).

Why does the verb måste come before the subject vi?

Swedish is a V2 (verb‐second) language. When you start a main clause with an adverbial like Den här gången, the finite verb must occupy slot two, so it inverts with the subject:
Adverbial (1) + Verb (2) + Subject → Den här gången måste vi…

Could I also say Vi måste vara mer organiserade den här gången?

Yes. Placing the time phrase at the end removes the need for inversion. You’ll have the normal Subject–Verb order:
Vi måste vara mer organiserade den här gången.

Why is it mer organiserade and not mer organiserad?

Organiserad is a past participle used predicatively, and Swedish adjectives agree in number and gender. Since vi is plural, you use the plural form with –e: organiserade.

Why do we use mer instead of adding -are to organiserad?

Short adjectives often form comparatives with -are (e.g. stor → större). Longer adjectives (usually three syllables or more, like organiserad) use mer:
mer organiserad (then agree in plural → mer organiserade).

Why is there no att before vara in måste vi vara?

After modal verbs (måste, kan, ska, etc.), Swedish drops the infinitive marker att. You say måste vara, not måste att vara.

What exactly is gången here, and why the -en ending?

Gång (“time” or “occasion”) is a noun. Gången is its definite singular form: “the time” or “the occasion.” Combined with den här, it means “this time,” so you get den här gången.

Could I replace den här with denna and say Denna gången?

You can use denna (“this”) in more formal or written language, but it must agree in gender and definiteness:
Denna gång (indefinite) or Denna gången (definite) is grammatically possible but less common in spoken Swedish. Most learners and speakers prefer den här gången.