Learn Hardcore German: Wir kündigen den alten Vertrag vorher, und nachher unterschreiben wir den neuen. - We cancel the old contract beforehand, and afterwards we sign the new one.

cancel/terminate a thing: den Vertrag kündigen (accusative)
give notice to a person: jemandem kündigen (dative), e.g., dem Mitarbeiter kündigen. So in Wir kündigen den alten Vertrag …, the contract is the direct object, hence accusative: den alten Vertrag.

Is the comma before und necessary?

No. With two main clauses joined by und/oder, the comma is optional. You can write either:

Wir kündigen … vorher und nachher unterschreiben wir …
Wir kündigen … vorher, und nachher unterschreiben wir … Use a comma if it helps readability; both are correct.

Why does the verb come after nachher in the second clause (… und nachher unterschreiben wir …)?

German main clauses are verb‑second (V2). If you put an adverb like nachher first, the finite verb must come second, and the subject moves behind it: Nachher unterschreiben wir …. Without fronting the adverb, you can keep normal order: … und wir unterschreiben nachher den neuen.

Can I omit the second wir?

Yes, if the subject is the same, you can leave it out in the second coordinated clause:
Wir kündigen den alten Vertrag vorher und unterschreiben nachher den neuen.
Here the verb can stand first in the second clause (after und), which is common in coordinated sequences: …, sah und siegte-style.

Is the placement of vorher at the end natural? Where else can it go?

It’s acceptable, but many would place a time adverb earlier. All of these are fine, with slight differences in emphasis:

Vorher kündigen wir den alten Vertrag …
Wir kündigen vorher den alten Vertrag …
Wir kündigen den alten Vertrag vorher … (more neutral/afterthought feel) Rule of thumb (Te-Ka-Mo-Lo): time elements tend to appear earlier in the middle field.

Could I use bevor/nachdem instead of vorher/nachher?

Yes, with subordinate clauses:

Bevor wir den neuen unterschreiben, kündigen wir den alten. (bevor = before)
Nachdem wir den alten Vertrag gekündigt haben, unterschreiben wir den neuen. (nachdem = after; verb to the end in the subordinate clause and perfect is typical) These versions sound very natural and tightly link the sequence.

Is nachher the same as danach, dann, or später?

Close, but nuances differ:

danach = “after that” (explicitly refers to the prior event; very common in writing).
dann = “then/next” (neutral sequence marker).
nachher = “afterwards/later on,” a bit more conversational.
später = “later (at some later time),” often less tightly tied to the just-mentioned action. All work here; a very natural rewrite is: Zuerst …, und danach/dann …

Why is it den neuen (lowercase) without repeating Vertrag? Should Neuen be capitalized?

Dropping the noun is common when it’s obvious from context: … unterschreiben wir den neuen [Vertrag]. In such ellipses, keeping the adjective lowercase is standard. Capitalizing (den Neuen) makes it a fully nominalized adjective and can suggest a different referent (often a person: “the new guy”), so lowercase is safer and more idiomatic here.

Why are the adjective endings -en (alten, neuen)?

Because with a definite article in masculine accusative singular (den), weak adjective endings are used: den alten Vertrag, den neuen (Vertrag). Pattern: der–die–das give the case/gender information, and the adjective takes -e in nominative singular (der alte) but -en in all oblique cases (den alten, dem alten, des alten).

Can I just say “den Vertrag beenden” instead of “kündigen”?

You can, but there’s a nuance. Kündigen is the standard legal/contractual term for giving notice to end a contract/subscription. Beenden is more general (“to end/finish”) and is fine, but sounds slightly less technical. Other options: aufkündigen/aufheben/auflösen (context-dependent), stornieren (orders/reservations, not contracts in general).

Does kündigen need a preposition (like von)?

No preposition is needed for the object: den Vertrag kündigen. If you want to name the provider/party, use bei: den Vertrag bei der Versicherung kündigen (“cancel the contract with the insurer”). With a person, you use dative: dem Mieter kündigen.

Is unterschreiben a separable verb? Why not “schreiben … unter”?

Unterschreiben is not separable. It’s a fixed verb meaning “to sign,” with participle unterschrieben (not “untergeschrieben”). A near-synonym is unterzeichnen (more formal).

Can I make it sound more idiomatic with zuerst/dann?

Yes, that’s very natural:

Zuerst kündigen wir den alten Vertrag, dann unterschreiben wir den neuen.
Erst kündigen wir den alten Vertrag, danach unterschreiben wir den neuen.

Why is present tense used for future plans?

German often uses the present for scheduled or near-future actions, especially with time/sequencing adverbs: Wir kündigen …, und nachher unterschreiben wir … The future tense (werden + infinitive) is possible but unnecessary here.

Why is Vertrag capitalized but alten/neuen not?

All nouns are capitalized in German, hence der Vertrag. alten/neuen are adjectives and stay lowercase unless they are fully nominalized (e.g., der Alte, das Neue). In this sentence, den neuen is an adjective with an omitted noun, so lowercase fits best.

Is früher a valid alternative to vorher here?

No. früher usually means “in the past/earlier (in time)” in a general sense, not “before that in the current sequence.” For step-by-step sequencing, use vorher, zuerst, dann/danach, nachher.