Learn German: Wir klettern rauf. - We are climbing up.

rauf is the shortened, colloquial form of either hinauf (up, away from the speaker) or herauf (up, toward the speaker). In everyday spoken German, people usually just say rauf instead of the longer forms.

If rauf replaces both hinauf and herauf, how do we know the direction relative to the speaker?

You don't! That is the convenience of the 'r-' words. By using rauf, you are just specifying the direction 'upwards'. The listener understands from context whether it is toward or away from the speaker, or it simply does not matter for the story.

Is rauf a separate word here, or is it part of a separable verb?

It functions here as the separable prefix of the verb raufklettern (to climb up). Because this is a standard main clause in the present tense, the base verb klettern is conjugated in the second position (Wir klettern), and the prefix rauf gets kicked to the very end of the sentence.

Could I write this exact sentence in a formal text?