"A person who represents himself has a fool for a lawyer." It's an old adage, but it's true. See if you can find a lawyer who'll represent you for a flat fee - say, one-third of the total. It won't be easy, because $10,000 isn't a lot for representation in contested litigation, but it's better than trying a case pro se. Chances are, your adversary will have a lawyer, and going up against a professional won't help your chances.
Where your dispute arose will determine whether you have to file in NY or California. If you can show that at least part of it took place in NY - if he sent you money in NY, or delivered something to NY, you can sue him in NY, but you'll have to serve him in CA. If you have to sue him in CA, you'll be at even more of a disadvantage. Any friends who are lawyers in NY won't be able to advise you. Your adversary can bury you in discovery demands in either place, and you'll need a lawyer. You don't have enough money in dispute to file in federal court.
Discovery is the process where each side has to disclose to the other (in a civil case) all of the evidence they have before trial. You can't hold anything back, unless it's material prepared specifically for trial (which is called "attorney work product" (and if you don't have an attorney I don't think a pro se can claim this exception, although I'm not sure), and privileged communications between an attorney and a client (which obviously won't apply if you're pro se). This means that you'll obviously have to turn over EVERYTHING that forms the basis of your case before you ever go to trial. Of course, your adversary will, too, if he's pro se; if he has counsel, he can claim the work-product and privilege exceptions.
Interrogatories, which are written questions that require written answers under oath (which I think are used in CA) are very onerous, but can be very helpful. They're used very often in NY. Each side sends a list of questions to the other, and so long as they can lead to the discovery of admissable evidence (even if the questions themselves aren't admissable), they're appropriate (but they have to be justifiable). Depositions (testimony under oath before a court reporter) are also permitted. The transcripts can be quite expensive, but they're usable in court, to impeach a witness if he testifies at trial contrary to his deposition testimony.
Document discovery is also used. This would include the emails. They have to be authenticated through testimony. They are not self-proving or self-authenticating. Certain records are - government documents, primarily. With other documents, you have to lay a testimonial foundation - that you prepared them on your computer, sent them to a particular email address that you knew belonged to the defendant, that only the defendant used that email address, how you learned that that was the defendant's email address, that you'd used that address for the defendant before the controversy arose, that he sent you emails from that address, that his emails were responsive to your emails, etc. That would lay the foundation for whatever particular emails you want to use as evidence. If you could testify that you talked to the defendant (in person or by phone) about the emails that would help. Obviously, all testimony has to be truthful.
Again, I urge you NOT to proceed pro se.
I agree with and endorse Susan Pernick's expanation and conclusion, and simply want to emphasize the futility of trying to handle this matter pro se. You'd probably be better off focusing your attention on other money-generating ventures than pursuing this on your own. Get a collection lawyer to handle it for you, either in NY or CA, as appropriate, depending on the facts. They'll likely charge you a contingent fee of between 25 and 33%, payable only on the sums they actually recover for you. In my opinion, the decision is a "no-brainer." And if you knew about litigation what litigators know about it, you'd likely agree with Susan (who I don't know) and me. Please note that I practice in NY (but don't do collection work). As they say, "Your Mileage May Vary." Best of luck.