Learn Dutch: Zwaaiend gaan we weg.

It comes from the verb zwaaien (to wave). To make it a present participle meaning "waving", you simply add a -d to the full infinitive. In this sentence, it acts as an adverb to describe how we are leaving.

Why is it gaan we instead of we gaan?

Because the sentence starts with Zwaaiend. In Dutch main clauses, the verb must always take the second position. Since Zwaaiend takes the first spot, the verb gaan comes second, and the subject we has to swap places and come right after it.

Why are gaan and weg separated?

They belong to the separable verb weggaan (to leave or go away). When you use a separable verb as the main verb in a present tense sentence, the conjugated part (gaan) goes in the second position, and the prefix (weg) gets kicked to the very end of the sentence.

Can I write this sentence without swapping the subject and verb?

Yes! If you start the sentence with the subject, the word order goes back to normal: We gaan zwaaiend weg. Moving Zwaaiend to the very front just puts a little more emphasis on the action of waving.