Learn English – “I’m going to take and stir the cake mix”

Traditional grammar defines the sentence as 'a complete thought containing a subject and a predicate'. So by this definition the second 'sentence' is incorrect, but of course writers constantly break the rules of grammar to achieve a particular effect.

The problem I see with the 'sentence' is that it may lead the reader astray. He or she gets to the end of the string of words (a long noun phrase) thinking that they comprise the subject, but then the predicate is missing. This will possibly cause a backtrack to reread the first sentence in order to make sense of the whole. It's generally inconsiderate of your readers to make them do this.

(And no, I would not recommend breaking this traditional rule in an application letter.)

In standard Modern English, negation is achieved by adding not after an auxiliary verb. If no such verb is present then the dummy auxiliary do (does, did) is introduced. For example:

I have gone → I have not gone (have is the auxiliary)
He goes → He does not go (no auxiliary in the original sentence)

Different rules apply in subjunctive, imperative and non-finite clauses.

So, here in your question, only the first sentence is correct, the second can be fine, depending on what you mean (but without the comma), and the third is incorrect.

I'm NOT going to Spain to have fun.

The correct form for the second sentence:

I'm going not to have fun, but to look for a job.

or with a slightly different meaning:

I'm not going to have fun, but to look for a job.

And the correct form for the third sentence:

I'm not going to have fun, but study instead.

or

I'm going not to have fun, but study instead.

You can keep this in your mind that not must always appear after the auxiliary and before the main verb.

See the Wikipedia article about negation in English for more information.