Learn Dutch: Mijn nek is stijf.

You would put wat right before the adjective it modifies. The sentence becomes Mijn nek is wat stijf. In this context, wat acts as an adverb meaning 'somewhat' or 'a bit', which is very common in everyday spoken Dutch.

I've also learned the word 'hals' for neck. Why is 'nek' used here?

In Dutch, nek specifically refers to the back of the neck, which is the part that gets stiff from muscle tension. Hals refers to the front of the neck or throat area, like where you wear a necklace or where you swallow.

If I want to say 'I have a stiff neck' instead of 'my neck is stiff', does the word 'stijf' change?

Yes. When placed directly in front of a noun, the adjective gets an -e ending. Because of Dutch spelling rules, the f also changes to a v, making it een stijve nek. In our example sentence, stijf comes after the verb is, so it keeps its base form without the -e.