Learn German: Wir räumen das Zimmer auf. - We are tidying up the room.

Because it belongs to the separable verb aufräumen (to tidy up). In German main clauses, the core part of the verb (räumen) takes the normal second position, while the prefix (auf) gets detached and pushed to the very end of the sentence.

If I add more words to this sentence, like "today", where does auf go?

It stays at the very end! For example, if you say "We are tidying up the room today," it becomes: Wir räumen das Zimmer heute auf. The separated prefix always acts like a bookend at the end of the clause.

How do I look this verb up in a dictionary?

You need to search for the full, unseparated infinitive (dictionary form): aufräumen. Dictionaries always list separable verbs with their prefixes attached to the front.

Why does the verb end in -en here (räumen)?

Because the subject is wir (we). Even though this is a separable verb, the core part conjugates exactly like a normal verb. The stem is räum-, and it takes the standard -en ending to match wir.

Does aufräumen mean doing a deep clean, like washing the floors?