Learn Dutch: Wij gaan vroeg slapen om uit te rusten. - We are going to sleep early to rest.

Uitrusten (to rest) is a separable verb. When you use te with a separable verb, it always gets sandwiched between the prefix (uit) and the main verb (rusten). When written down, they combine into a single word: uit te rusten.

Could I leave out om and just say Wij gaan vroeg slapen uit te rusten?

No, to express a goal or purpose (meaning "in order to"), you must use the combination of om and te. The word om introduces the purpose, and te goes right before the infinitive verb at the end.

Why doesn't the verb slapen go at the very end of the entire sentence?

In the main clause, the second verb (slapen) goes to the end of its own section (Wij gaan vroeg slapen). The om ... te phrase acts as a separate block added afterward to explain why you are going to sleep. It doesn't push the main clause's verbs any further back.

Does gaan slapen mean "we will sleep" or "we are going to bed"?

It effectively means "we are going to bed". While the verb gaan (to go) combined with another verb is often used to show a future plan, gaan slapen is specifically the standard Dutch phrase for going to bed or heading to sleep.