How far should I be with writing a story before looking for a publisher, or do I have to finish the book first then look for a publisher?

You can always start researching agents / publishers, but your novel has to be finished before you can start submitting to them. Basically it has to be as perfect as you can possibly get it. I have a novel that just started making the rounds last week, and here's the amount of work that went into it before it reached that point:

- first draft

- second draft

- proofread by myself

- three beta-reads

- revision

- proofread by myself

- proofread by proofreader

The competition is fierce and agents and publishers can pick and choose among the very best, so you want to do whatever you can to stack the odds in your favour.

You have to finish it first.

It should be as close to being ready for publication as you can make it - complete, edited and polished.

Given the length of time that publishers and literary agents sometimes take to reach a decision about whether they're interested in your manuscript, it's tempting to write a few chapters, query them with those, and write the rest only if they're interested. The trouble is they receive hundreds of manuscripts for every one that they offer to publish or represent, so they can afford to be picky. When they express an interest in one, they're accustomed to receiving the full manuscript right away, or within a day or two at the most. If you try to stall them while you frantically start writing the rest of it, then eventually have to admit that the rest doesn't exist, you've just blown your chances with that publisher or agent - not only for this manuscript, but possibly forever.

Three days before you start.

Why the heck did you not learn about the BUSINESS of publishing before you took the time to write?

If your book content is good and your agent thinks it will sell, they will help you proof read and fully edit it. Be prepared to make changes they suggest, especially to plot. That means you definitely need to have the whole thing written first.

You should have written it completely, re-written it a few times, proof-read it, and then had it proof-read by someone whose English is extremely advanced.

You should also have started on your next novel and at the very least have a rough outline of the entire story read.

And you don't look for a publisher yourself anyway.

You need a literary agent for that.