I agree with everyone else. I started pro-se and it was so incredibly overwhelming, I had absolutely NO IDEA what I was doing and didn't have the time to be going to the courthouse/legal library to figure it out every time a piece of paper was filed. Plenty of people do successfully represent themselves though, even with no legal knowledge so I'm not trying to say it can't be done! If you *do* decide to go pro-se (you can always hire a lawyer later on), check with your county courthouse, they should have a free family law pro-se program or will be able to point you to an organization that can help. This is not the same as pro-bono legal counsel, pro-se programs typically just help you with making sure you filled out your paperwork and are filing the right forms.
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And I have to very strongly second everyone who said to get a consult. And get more than one! They're usually free and will be able to give you an accurate answer about your initial question about who gets a lawyer/files first. Also for pro-se, it can be extremely helpful to have started a relationship with a lawyer if you run into any difficulty with your case and just need to pay for an hour or two of legal counsel.