Who should an author approach first? Editor? Agent? Publisher?

Depends. If you're looking for traditional publishing, make your manuscript as good as you can on your own and then query an agent. I personally wouldn't spend thousands on an editor before finding an agent.

Agents get paid by finding you a publisher and taking a cut of your royalties so if you have already found a publisher then you don't need an agent and an agent won't want you because they can't get paid if you've already signed a contract. However a lot of the bigger publishing houses won't talk directly to authors and make you go through an agent. Therefore an agent is a good idea.

If you get a traditional publishing deal, they'll pay for the editing.

Of course, none of this applies if you're going with self publishing. With self publishing, you want to get an editor and you really don't need the agent in that scenario.
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Personally I would say it should be an editor first, make sure that book is in mint condition so that the publisher has no reason to turn it down - they do so for the flimsiest of reasons - send the first 5000 words - you miscount and send 5002 and it's a no ... in my experience (sadly people have said that. I get so angry over that because different apps give different word counts. :( Once the book is in the best condition it can be, then approach an agent ... that's just my view though.
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If the author is self-publishing, the author should approach an editor and forget the other two. If the author is seeking traditional publishing, the answer will depend on the quality of the manuscript and the requirements of the publishers. Many take only authors who have agents, and some take authors who don't have agents. It all depends on the situation and several factors.
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I do 18-22 pass throughs - sadly, for reasons I won't go into on Facebook, I am unable to look for all issues at the same time. I am always happy to share my list of what to look for. Only once I've done the edits that I can deal with do i approach my favourite editor in the word (but you can go with whomever you like), Erica Farner. I hope that by getting my books as well edited as possible, her job will be made easier. However, as I said in a post earlier today, not everybody has the same skills level - especially those for whom English is their second (or even third) language. In that case, an editor is even more essential. They can help with every area - I once approached a company and they responded with, "although this manuscript stands head and shoulders above many other submissions, you need an editor to point out a few plot points which you've muddled up." Hopefully Erica hasn't spotted any terrible ones ...
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You can read your manuscript 30 times and I guarantee you'll start reading what you think is written instead of what ACTUALLY is written. As a writer, editor and proofreader, I strongly recommend finding a good editor, not a friend who will only tell you good things. You want your manuscript as good as it can be prior to trying to submit it to agents and publishers and you could find one or both of them very quickly. So (and yes, I am biased, but with a good reason), you need an experienced editor as your first port of call. If you're lucky, you may find one who does editing AND proofreading (such as me). I hope this all makes sense and I wish you good luck.
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Editor, so you can create the perfect product. Then you can approach the publishers on your own (be sure to follow their rules on their website!). If you get rejected a lot, contact an agent and work something out, so THEY can work directly with publishers and you're more likely to get a deal.