It depends on the country, so you should check with your local administrative unit. Here in Slovenia, I needed an apostilled birth certificate from my state before I could get married here and an apostilled fbi background check for my residence permit (if you want to stay more than 90 days). If you need those, it is definetely eaiser to get them while in the states. You can visit the U.S together in the meantime with ESTA, but be prepared to show proof that you will return back to EU, e.g. return tickets, employment contract.