Learn Hardcore French: Le train arrive enfin, et nous y montons rapidement. - The train finally arrives, and we get on it quickly.

enfin means finally or at last. When used to modify a verb in French, it can come after the verb without changing the meaning:
Le train arrive enfin = The train finally arrives
You could also put it at the beginning (Enfin, le train arrive), but that shifts the emphasis slightly.

Why is there a comma before et?

In French, a comma can separate two independent clauses, especially when the subject changes. Here the first clause subject is le train, the second is nous, so the comma before et marks the pause between them.

Why do we need nous again in the second part?

Because the subject changes from le train (3rd person) to nous (1st person). French requires an explicit subject pronoun for each finite verb, so you must repeat nous for clarity.

In nous y montons, what does y replace?

y is a pronoun of place that replaces dans le train. It always goes before the verb in simple tenses, so nous y montons literally means we get on it.

Why isn’t monter followed by dans or sur?

Normally you say monter dans le train (to get on the train). Here dans le train has been replaced by y, so there is no need for the preposition anymore. You still use monter intransitively as a movement verb.

What’s the difference between rapidement and vite?

Both are adverbs of manner meaning quickly, but vite is more colloquial and shorter, while rapidement sounds more formal or precise. They follow the same placement rules (usually after the verb), for example: nous y montons vite / nous y montons rapidement.

How would you say we got on the train in the passé composé?

Because monter is an intransitive movement verb, you use être as the auxiliary: nous y sommes montés. Notice agreement: if nous refers to a mixed-gender or masculine group, add -s (montés); if a feminine group, montées.